Your posts are fantastic, Leo. As far as I'm concerned that's a novel proof of \(f(f(x)) = -x\) can't be real valued. We have a couple of these problems on these forums. I hope you post more. You have a great brain.
To iterate \(\text{conj(z)} = z^* = \overline{z}\) you need to do Grothendiek level shit. Iterates only exist in exotic metric spaces, that are probably better represented using p-adics, or something like that. It'll never be analytic/continuous/differentiable. Nothing good is to come from looking at it like that. But as an abstract algebraic idea, it does exist. Assuming the axiom of choice there are uncountably many automorphisms of \(\mathbb{C}\to \mathbb{C}\) as a field. But the only continuous one is conjugation and the identity mapping. Not to rain on your parade, but this question is pretty much settled. I admire your spirit. Love your energy.
Regards, James
To iterate \(\text{conj(z)} = z^* = \overline{z}\) you need to do Grothendiek level shit. Iterates only exist in exotic metric spaces, that are probably better represented using p-adics, or something like that. It'll never be analytic/continuous/differentiable. Nothing good is to come from looking at it like that. But as an abstract algebraic idea, it does exist. Assuming the axiom of choice there are uncountably many automorphisms of \(\mathbb{C}\to \mathbb{C}\) as a field. But the only continuous one is conjugation and the identity mapping. Not to rain on your parade, but this question is pretty much settled. I admire your spirit. Love your energy.
Regards, James