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It makes people compare them to unicorns and magic. But they’re not a storyteller’s plot device. 
They’re really just a different category of number. They have practical uses in the real world. They got stuck with an unfortunate name, and mathematicians have been explaining the resultant misconceptions for centuries
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Mathematician here

When we say number, we usually mean amount--or a concept to represent an amount, if you're less Platonist. But of course, the numbers called imaginary do not fit such a requirement

When I say "number" I mean "something that you can add or multiply to other numbers". Complex numbers certainly qualify. However, the word "number" is a very vague, ambiguous term, and mathematicians have much more precise terms for collections of "numbers" or "things" that act more or less like numbers. If you ask me "are quaternions numbers?" I'd say "if you like."

No amount can be squared to equal any negative number

This kind of begs the question - what do you mean by an "amount" ? If I have an amount i of something, that amount can certainly be squared to produce -1... And before you say "you can't have i of something", note that that's a very broad statement about all possible quantities everywhere, and I doubt you've done a careful survey of all possible things.
For example, if I have an AC current flowing through a circuit, and want to measure the "amount" of resistance of a component, any electrical engineer will tell you "the correct word is 'impedance', not resistance, and yes, the amount of impedance can be a complex number".
If you ask a quantum phsyicist to describe the "amount" of "probability wave" passing through space at some point, that amount will also be a complex number.

A common argument is that "real numbers are no more imaginary than imaginary numbers"

Indeed. We can have an amount "3" or "4" of apples, say, but we can never have sqrt(2) of an apple. No matter how much apple we have, it will never be sqrt(2), nor any other real number, since there's always some fundamental uncertainty in how much of something there is. We can never really have a curve that is pi times the length of a given straight line. We can in an abstract theoretical sense, but not in reality. pi is never an "amount" or length or mass or whatever, since "amounts" always have built-in uncertainty. >


"you can't have i of something", note that that's a very broad statement about all possible quantities everywhere, and I doubt you've done a careful survey of all possible things.

I have not, however, as amounts overall have their own universal nature, I can logically assume that nothing can exist to the extent of i.

When I say "number" I mean "something that you can add or multiply to 
other numbers". Complex numbers certainly qualify. However, the word 
"number" is a very vague, ambiguous definition, and mathematicians have 
much more precise terms for collections of "numbers" or "things" that 
act more or less like "numbers"

Given dictionary defintions, I would say that most people would understand the term differently.

Indeed. We can have an amount "3" or "4" of apples, say, but we can 
never have sqrt(2) of an apple. No matter how much apple we have, it 
will never be sqrt(2), nor any other real number, since there's always 
some fundamental uncertainty in how much of something there is. We can 
never really have a curve that 
is pi times the length of a given straight line. We can in an abstract 
theoretical sense, but not in reality. pi is never an "amount" or length
 or mass or whatever, since "amounts" always have built-in uncertainty.

Wouldn't it depend on what you mean by uncertainty? If you mean uncertainty to a sentient being, than yes, but there is still an objective amount, is there not? >


I have not, however, as amounts overall have their own universal nature

I would invite you to define that "universal nature", maybe define exactly what you mean by the term "an amount". For example, what about my electrical engineering example? The impedance of a circuit component is certainly something we can measure, why would it not be an "amount"? 

Wouldn't it depend on what you mean by uncertainty? If you mean uncertainty to a sentient being, than yes, but there is still an objective amount, is there not?

The uncertainty the universe presents us with is more fundamental than that. At the deepest level of physical reality, it's impossible to measure location (and hence length) precisely, without sacrificing precision about movement. It's impossible to measure energy (and hence mass) perfectly precisely unless one has an infinite amount of time. Every physical quantity you might call an "amount" has this intrinsic uncertainty built in at the fundamental level. There's no "objective amount" hidden underneath. >


I would invite you to define that "universal nature", maybe define exactly what you mean by the term "an amount".

The former I can do, the latter I can not. That would be like trying to define space or time. which I cannot do, as it has not been logically presented to me, but you already know, as you live in space and time.
But here's the universal nature of amounts:
All amounts are existent from one infinitesimal point to another(unless infinite, in which case existence is throughout the universe). Individual things exist based on something distinguishing them from other things. Thus, if there are many distinguishing properties leading to different individuals, individuality is subjective, but not if there is only 1. The individual forms the basis of our number system
​
Edit: I can define the TERM"amount" but not the concept it represents. But here's the term's definition: quantity. >


All amounts are existent from one infinitesimal point to another(unless infinite, in which case existence is throughout the universe). Individual things exist based on something distinguishing them from other things. Thus, if there are many distinguishing properties leading to different individuals, individuality is subjective, but not if there is only 1. The individual forms the basis of our number system

That doesn't seem to exclude complex numbers. >

Complex numbers extend from one place to another? But then why can't they be compared with non-complex numbers. >

Complex numbers are numbers with a real component and an imaginary component. They're actually quite comparable with non-complex numbers, insofar as you mean real numbers or imaginary numbers. The real portion has all of the "concreteness" of the real numbers. The imaginary portion is a quantity perpendicular (it adds a second dimension) to the real portion. You can conceptualize the set of all complex numbers as a plane in the same way you can conceptualize the set of all real numbers as a line. >

Okay, so what do the two axes of the plane represent? >

Pardon? My answer to your question is basically going to be a repeat of what I said above. Maybe it will help if I express it with symbols?
Consider a real number, a, and an imaginary number bi. A complex number, z, is expressed as z = a + bi.
The first axis is the real axis. The second axis is the imaginary axis. Both are so named because of the names we give to the sets each number belongs to. There are many consequences that come from this. One thing we may be able to immediately recognize that a real number is just a complex number with imaginary part bi=0. Similarly, an imaginary number is a complex number with real part a=0.
EDIT: You might not find this interesting, but one fun thing I like about complex numbers is you can find numbers u and v such that u + v = 8 and have u * v equal any number you want. For example, 116. The specific numbers I've chosen here don't matter, but I did pick them because they are simple enough to have pretty solutions. >

Okay, but can you really compare complex numbers to real numbers? Think about a one dimensional sentient creature. They would not be able to know that 2 other spatial dimesnions exist, that's inconcievable. SImilarly, how do we know that there is an imaginary dimension, when we only see our real dimension(yes, we us i, but seemingly only as a tool as a hypothetical base. >

What does it mean for a number to exist? what makes positive integers "real" as compared to complex numbers? I mean you also can't have "pi" of something or "e" of something so are transcendental numbers also imaginary? Hell even negative numbers, you can't really have -1 of something so is that imaginary? >

This is gonna be cliche, but I've never seen i of something. I've seen individual things before. Also, can you really not have e or pi of anything, or is it just infinitely unlikely, given the uncountable infinity of real numbers. And negative numbers refer to how much 0 there is, positive numbers refer to how much one there is. >

I mean, the world is mostly discrete, like if you get down to there's not an infinite number of things, there's only so many quarks and other fundamental particles so you can't divide into anything that isn't a rational number at the end of the day.   
And what does "negative number refer to how much 0 there is" even mean? I have no idea how to even parse that >

​

I mean, the world is mostly discrete, like if you get down to there's 
not an infinite number of things, there's only so many quarks and other 
fundamental particles so you can't divide into anything that isn't a 
rational number at the end of the day.

You could say there's infinitesimals. After all, for something to stop existing and start not existing, there has to be an infinitesimal end. >

So infinitesimals exist? you've seen those?
And

After all, for something to stop existing and start not existing, there has to be an infinitesimal end.

what does this mean? why does there have to be an infinitesimal end? What even is an infinitesimal end? I also note you didn't answer my question about what your negative numbers quote meant >

Do we both agree that there are 3 dimensions of space? >

The concept of numbers, whether imaginary or not extend far beyond what we might encounter on an everyday experience. 
For example most quantities we use on an everyday level will be natural numbers, (1,2,3,4, etc.) such as things we count like an apple.or two apples.  Or say ratios of natural numbers (example 5/3, 1/2); I have two times as many apples as you.  A decimal such as 1.2 can be thought of as the ratio 12:10.
But what about negative numbers, zero, and irrational numbers? We never encounter these in our everyday life. What does it mean to have negative apples? Ludicrous! How can I count nothing of something? I certainly can't ever measure sqrt(2). But yet these are all real numbers and people seem perfectly comfortable with them.  (In fact i would  argue that you never measure a number ever; rather an uncertainty; also things in real life have units which can be represented by any number depending on how i define units. If someone told me they measured "five" in my lab it would have no physical meaning whatsoever). So I would argue that the majority of real numbers are NOT something we could ever measure, and so claiming real numbers are grounded in our everyday day physics life is quite wrong IMK (Fun fact, the natural numbers are "smaller" in quantity than the irrational)
I  suspect that our comfortableness with these real numbers is that   we were taught that these concepts are "normal"  and "unsurprising". But these concepts are quite abstract and we are just accustomed to using them. (The history of math is quite interesting in fact). Math becomes so much more interesting when we make it abstract and leave the world of arithmetic (ie. 1+1 =2)
And so this arrives at the word "imaginary number". To me "imaginary " things aren't as important as "real" things. But as you can see real numbers are in fact quite abstract, and I argue that adding sqrt(-1) is not really that big of a leap.  You are just generalizing the rules of real numbers (namely the sqrt) and adding an extra symbol for shorthand "i". It's not a perfect analogy, but it's like adding "0" to the set of natural numbers. I'm sure someone before the invention of zero would be very upset at us adding this new symbol which seems not based in "reality"
And so calling them imaginary relegates "i" to something exotic or not important, where the opposite is very true. So much of the world can be described more elegantly using complex numbers. It's by no means guaranteed but we tend to think that elegant and simple explanations are closer to "Truth" rather than complicated and messy explanations. If my students appreciated imaginary numbers better, they would have a mich deeper (and  I would argue better) understanding of physics and math.  Like this is one of my top ten biggest gripes about my undergrads and math (the importance of calculus is way over emphasized to the general public). Seriously, the importance of imaginary numbers and the fact that quantum is probably the most important physics for ALL modern technology is SO underappreciated.
Math and numbers are abstract, but super useful in explaining the universe. Why should we bias ourselves to numbers that we experience on an everyday experience? The rules of the universe don't care about our limited experiences. If the universe is best explained with complex numbers (which have real and imaginary parts) then it is my biases which are wrong.
(My experience : phd in physics with emphasis  on laboratory quantum  mechanics and now postdoctoral researcher. Math is not my strong suit, but more than qualified for this discussion)
Some examples where using complex /imaginary numbers is more elegant and useful than real numbers alone :

anything that is oscillatory (sound waves, music,.quantum phase of a wave functions, electromagnetic waves (ie light, telecommunications), alternating circuits; any wave really. This is because Euler 's law says exp(ix).= Cos(x) + isin(x)
generalization of sqrt, such as cube root, or higher order
things that use angles (depends on context)

Also, I saw you comment that in quantum you measure probability and I want to point out that according to quantum mechanics , the complex wave function is the fundamental physical object. You could measure the same probability but have different quantum states. Also Schrodinger's equation which is the equivalent of newton's laws (F=ma) for quantum mechanics  requires the use of imaginary numbers. In other words you could not describe how a quantum states evolves in time without invoking complex numbers.
TLDR : Real numbers are already not well related to everyday life and are super abstract. Yet they and imaginary numbers are critical in describing how the universe behaves, and that grounds them in reality, which can be against our intuitive experiences. Also, history has shown that advances in math which may seem abstract and non applicable initially have very real (pun intended) and measurable consequences on our understanding of the universe. Also also, humans are dumb and we shouldn't be surprised that our intuition about math and it's physicality don't agree with what our universe obeys.
(It's actually pretty awesome that math, which is based on logic that we humans invented is so remarkably good at physics to describe the universe. Thanks goodness!!) >

We don't encounter zero, but there are things we don't encounter which exist, so 0. >

Take the 2D Cartesian plane and define two operations by (a, b)+(c, d) = (a+c, b+d) and (a, b)(c, d) = (ac-bd, ad+bc). If you reinterpret the points as vectors, the former operation corresponds to vector addition and the latter corresponds to multiplying that magnitudes of the vectors and adding up the angles they make with the positive x-semiaxis. Everything here is completely defined in concrete and visual terms. You can draw this and demonstrate various examples of the operations at work. No imagining needed if you can demonstrate it on graph paper. 
But what we have is exactly the structure of complex numbers (in fact this is one of the various well-known constructions of the complex numbers). We can relabel "a = (a, 0)" and "b * i = (0, b)" and we obtain exactly the notation of the usual complex numbers. 
So when you said, 

Therefore, nothing can be correctly referred to as existing to the extent of i*n, regardless of any unit of measurement. Something can only be referred to as existing to the extent i^n.

Maybe not as an amount, but if complex numbers are directions with magnitudes on a 2D plane then things can be correctly referred to as existing to the extent of i*n (arrows are an example) >

Complex numbers are directions on a 2d plane, but it seems that this plane itself is supposed to represent something. After all, aren't all numbers supposed to be amounts? >

Amounts, as in a magnitude. If you just have a magnitude, you're looking at a scalar. Numbers can also simultaneously communicate magnitude and direction. This is known as a vector. I can tell you to go thirty miles south and ten miles east with the number 10 - 30i. Of course, it is the application that decides whether the number you're talking about is a vector or a scalar. >

To me, imaginary means not concrete but rather an abstract concept. Infinity is abstract, for example. The integer “2” is concrete. >

Exactly(though I'm not sure I'd agree on infinity, but I digress), so since I believe i is abstract, I am okay with the term and think it makes sense. Do you contest that i is abstract? >

What about -2 is that concrete given you can’t have -2 apples? If you start with 2 apples and take 4 of them away, well you can’t! You can’t take 4 apples away if you only have 2 apples.
However I don’t think that -2 so mysterious. It’s just an extension or generalisation from natural numbers and makes things add up (literally). For example if a and b are simple counts then we can define a number
c = a - b
And with it:
a = c + b
b = a - c
Negative numbers allow us to do this. My point in raising negative numbers is to argue they are an extension/generalisation too in the same way as imaginary numbers. If you don’t consider -2 to be so mysterious (I do not) then you shouldn’t consider i to be that mysterious either.
In your terms that means do you consider-2 to be abstract? I think it’s either the case that both -2 and i are abstract (imaginary) or they are both not. >

-2 apples means 2 negative apples, whatever the opposite of an apple is. If you owe $20, you have -$20, because owing is negative having. >

But there is no such thing in the world as a “negative apple”. There is the concept of a “negative apple” which we know the rules for in apple arithmetic but that’s it. It is the fact this is a concept and not a real thing either. We can’t point to a collection of -1 apples any more than we can point to a collection of i apples.
FWIW I think it’s more an anti-apple than the opposite of an apple as when they’re brought together the apple is “annihilated” but we’re digressing. >

There is, it's the lack of an apple. >

I wish academics put more though into choosing terminology. Having overlapping definitions of so many terms in different fields is confusing. If we were to go with Gauss' idea of calling them lateral numbers, we would know right away there's something spatial involved but the layman doesn't make the mistake of thinking they know what they are. It's a unique term for a specific and decidedly advanced concept, so we shouldn't focus on relatability and clever metaphor through word choice, we should simply choose to be less ambiguous. >

How is "imaginary" ambiguous?