If one accepts the Axiom of Choice, then the automorphism group of $\mathbb{C}$ is a huge and wild group, very poorly understood.

But apparently if one does not accept the Axiom of Choice, then the automorphism group of $\mathbb{C}$ has size $2$, only consisting of the identity and complex conjugation.

What are other interesting (classes of) fields where similar things can be said about the automorphism group ? (That is, upon not accepting the Axiom of Choice, one ends up with an "easy" or even trivial automorphism group.)