A ring endomorphism or automorphism of the complex numbers is said to be wild if it is different from the identity or complex conjugation. In this answer it is said that “there are lots of [ring] endomorphisms [of $\mathbb{C}$] that are not automorphisms.”
If a wild endomorphism of the complex numbers exists, it must be non-continuous, for the only continuous endomorphisms are the identity and complex-conjugation. (Ring endomorphisms must be always $\mathbb{Q}$-linear and preserve the roots of the polynomial $x^2+1$. On the other hand, continuity gives us $\mathbb{R}$-linearity.)
My question is: how can we produce a non-bijective wild endomorphism of $\mathbb{C}$? The only mention of such a thing I've been able to found on MSE or MO is the answer I linked above. On the other hand, the papers by Kestelman and Yale don't mention this.
Here they explain a recipe to construct wild endomorphisms of $\mathbb{C}$. However, I don't understand the last step: “$g$ can be extended to a field endomorphism $f:\mathbb{C}\to\mathbb{C}$.” Specifically, I don't understand:
Why can we extend $g$ to a ring endomorphism of $\mathbb{C}$? (I don't know much field theory so probably there is some result that is being used and that I am unaware of.)
How can we guarantee that $f$ will be a non-bijective wild endomorphism?
