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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.
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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.