Field homomorphisms from $\mathbb{C}$ to $\R$

Assume that such a $\varphi:\mathbb{C}\rightarrow \R$ exists. Then, by definition of a field homomorphism,
$$\varphi(-1)=\varphi(i^{2})=\varphi(i)^{2}$$but $\varphi(-1)=-\varphi(1)=-1$. We therefore obtained that the real number $\varphi(i)$ verifies $\varphi(i)^{2}=-1$, which is impossible since the square of any real number is non-negative.

So no such field homomorphism can exist.