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