Does there exist a field of characteristic 0 that has no automorphisms other than the identity but has endomorphisms other than the identity? In characteristic $p$, one can take the function field of a curve with no automorphisms and then Frobenius would define an endomorphism but I am not sure about characteristic 0.
I do not know many interesting ways of producing fields, pretty much only algebraic closure and completion. Algebraically closed fields always have automorphisms so they are not going to work [1]. The completions of $\mathbb{Q}$ at non-trivial places have no endomorphisms so they are not going to work either [2].