There is the more general fact that any automorphism of any subfield of $\mathbb{C}$ can be extended to an automorphism of $\mathbb{C}$. For a proof, see the paper _Automorphisms of the Complex Numbers_ by Paul Yale of Pomona College. Here is a [JSTOR link](http://www.jstor.org/discover/10.2307/2689301?uid=3739792&uid=2129&uid=2&uid=70&uid=4&uid=3739256&sid=55994902273). In general, if $k$ is an arbitrary (EDIT: algebraically closed) field, my guess would be Yale' argument could be easily extended to show that any automorphism of a subfield $h\subset k$ can be extended to an automorphism of $k$.