Simon Thomas asked in
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 whether (under appropriate large cardinal assumptions) there is such an automorphism in $L(\mathbb R)[\mathcal U]$ where $\mathcal U$ is a nonprincipal ultrafilter on $\mathbb N$.