5:23 AM
This question was for a very long time at the top of unanswered questions:
163

It is well-known that it is consistent with $ZF$ that the only automorphisms of the complex field $\mathbb{C}$ are the identity map and complex conjugation. For example, we have that $\vert\operatorname{Aut}(\mathbb{C})| = 2$ in $L(\mathbb{R})$. But suppose that we are given a nonprincipal ultraf...

It was also explicitly mentioned in this meta thread:
14

Simon Thomas asked in Ultrafilters and automorphisms of the complex field whether the existence of non-principal ultrafilters (over the natural numbers) suffices to imply the existence of a nontrivial automorphism of the complex field $\mathbb C$. In set theoretic terms, the question is whe...

Recently, an answer was posted and it is not accepted. (I do not know enough to be able to judge the correctness of the answer.)
8

It seems not. It was shown by Di Prisco and Todorcevic (and reproved later by at least three sets of authors) that if sufficiently large cardinals exist (e.g., a proper class of Woodin cardinals), then after forcing with $\mathcal{P}(\omega)/\mathrm{Fin}$ (the infinite subsets of $\omega$, orde...