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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.
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