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.)

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.)