Are there any nontrivial differentiable automorphism of the complex numbers? I know there are many automorphisms, but I could only find one article that discussed them. I didn't read the entire thing, but it mentioned that AC is often necessary to construct them, but I didn't see whether it said that was always the case. Obviously, there's the identity; and then there's the conjugate which is trivial and continuous but of course not differentiable. My intuition is that any others would have to be pretty crazy, possibly not even integrable, but I haven't been able to prove that. Any insights?