I mean, yes of course i is an element in C, because it's a monic polynomial in i.

There's no "intend to". The complex numbers are what they are regardless of us; this isn't quantum mechanics where the presence of an observer somehow changes things.

It's not about observers, but about mathematical structure and meaning. Without answering the questions, you are being ambiguous as to what the structure of C is. For example, if a particular copy of R is fixed as a subfield, then there are only two automorphisms---the trivial automorphism and complex conjugation, since any automorphism fixing the copy of R would have to be the identity on those reals and thus the rest of it is determined by whether i is fixed or sent to -i. Meanwhile, if you don't fix a particular R subfield, then there is a vast space of further wild automorphisms. So this choice of structure---that is, the answer to the questions I posed---has huge consequences on the automorphism group of your conception. You can't just ignore it and refuse to say what the structure is.