Nope. Take any true mathematical sentence and (consistently) replace i with -i and it remains true; that is not the case for 1 and -1. Im(x) = 1 is meaningless; it would have to be Im(x) = ±1. (In fact you'll only ever see complex numbers in the form of a±bi, never a+bi alone.)
It's why you can't say e.g. -i < i; the signs on purely imaginary numbers are not an ordering.
Seems pretty true with 1 and -1. Map R with f(x) = -x and f(x) = |x - 1| for x in your new mapping is indistinguishable from f(x) = |x + 1| in R.
In any case I’d say this is arbitrary like using + for addition and - for subtraction. It seems like you’re just talking about the symbols themselves. I’m not sure how you get to half plane from there.
Sure, but I'm not sure I'm understanding the argument. I don't understand how a function like f(z) = e^z has to be symmetric about the real number line or how i and -i aren't distinguishable with something like Im(z) > 0. Is there a proof somewhere I can read?
It falls out of complex numbers satisfying the conditions of a field though I don't know of a specific "proof" of that (you generally don't "prove" definitions). You could equally say "i is indistinguishable from 1/i" or "i's additive inverse is its multiplicative inverse"; in either case it's an arbitrary choice which of the conjugates is positive and which is negative. The key being that you cannot say "i > -i" because of that.
