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There are ways to build C that result in:

1) Exactly one C

2) Exactly two isomorphic Cs

3) Infinitely many isomorphic Cs

It's not really the question of whether i and -i are the same or not. It's the question of whether this question arises at all and in which form.

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The question is meaningless because isomorphic structures should be considered identical. A=A. Unless you happen to be studying the isomorphisms themselves in some broader context, in which case how the structures are identical matters. (For example, the fact that in any expression you can freely switch i with -i is a meaningful claim about how you might work with the complex numbers.)

PP meant automorphisms, which is what the OP article is about.

Homotopy type theory was invented to address this notion of equivalence (eg, under isomorphism) being equivalent to identity; but there’s not a general consensus around the topic — and different formalisms address equivalence versus identity in varied ways.

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