[cperciva]

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.

[FillMaths]

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.

[cperciva]

You're assuming there has to be a "meaning". There isn't. We're just manipulating meaningless symbols.