Ring homomorphism $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$

Question

Exercise

Find all the ring homomorphisms $f: \mathbb C \to \mathbb C$ such that $f(\mathbb R) \subset \mathbb R$

My attempt at a solution

In a previous problem I've already proved that the only ring morphism $h:\mathbb R \to \mathbb R$ is the identity.

Here, we have $f|_{\mathbb R}=Id$. Take $z \in \mathbb C$ with $z=a+ib$, then $$f(z)=f(a+ib)$$$$=f(a)+f(ib)$$$$=a+f(i)f(b)$$$$=a+f(i)b$$

So, I need to determine the value $f$ takes at $i$. I am not so sure if $f(i)$ can take any value in $\mathbb C$ or if it has some sort of restriction in order to respect the structure of the ring morphism. I would appreciate some suggetions to finish the problem.

Answer 1

Hint: You'd need $f(i)^2=f(i^2)=f(-1)=-1$, at minimum.