If one accepts the Axiom of Choice (AC), 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. I should be more precise here (see the comments below): it is *consistent* with this assumption that the automorphism group of $\mathbb{C}$ has size $2$ (i. e., we can find models of ZF without AC in which this group has size $2$).

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 consistency of an "easy" or even trivial automorphism group.)

consistentwith ZF that every automorphism of a Polish group is continuous, in that case only the identity and conjugations are automorphisms of $\Bbb C$. $\endgroup$ – Asaf Karagila♦ Oct 23 '20 at 13:043more comments