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.

Ah, gotcha. I think I wasn’t understanding exactly what was being said. Thanks for the explanation.

I interpreted your words as "the complex plane has topology of a plane where conjugates are glued together".

Ah, sorry, no; it is a half plane in the sense that which way is "up" is completely arbitrary