If an endomorphism of $\mathbb{C}^*$ is assumed to be holomorphic, then it is not difficult to show that it must be of the form $z^n$ for some $n\in \mathbb Z$. I was wondering if this is true without any topological or analytic assumptions on the endomorphism. My intuition says that there should be lots of them, but I'm usually working with endomorphisms which are at least holomorphic if not a true morphism of varieties, so I'm not sure.

There are many "wild" automorphisms of the complex numbers, that preserve addition and multiplication, but not much else. These are hard to write down, however, as their existence seems to rely on the axiom of choice. See e.g. the great expository paper "Automorphisms of the complex numbers" by P. Yale, Mathematics Magazine 39(3), 1966. 

