TITLE: Wild automorphisms of the complex numbers
QUESTION [57 upvotes]: I read about so called "wild" automorphisms of the field of complex numbers (i.e. not the identity nor the complex conjugation). I suppose they must be rather weird and I wonder whether someone could explain in the simplest possible way (please) how I could imagine such wild automorphisms.
E.g. I suppose they are completely discontinuous.
E.g. are the real rational numbers fixed or any other set of complex numbers?
Can such an automorphism be pictured in a model?

REPLY [23 votes]: The complex numbers can be characterized algebraically as the unique algebraically closed field of transcendence degree continuum over $\mathbb{Q}$. I think the key to understanding automorphisms lies in thinking of $\mathbb{C}$ in this way and forgetting any other structure like topology or embedding of $\mathbb{R}$.
There are also lots of "good" subfields of $\mathbb{C}$, like, say, algebraic extensions of $\mathbb{Q}$, and any automorphism of a subfield can be extended to an automorphism of $\mathbb{C}$ (this requires AC, of course). Moreover, in fact there are lots of endomorphisms that are not automorphisms, and lots of subfields of $\mathbb{C}$ that are isomorphic to $\mathbb{C}$.
All of them are quite "bad" analytically - in particular, not Lebesgue measurable (because even any measurable group homomorphism $\mathbb{C} \to \mathbb{C}$ is automatically continuous).