$\R$-automorphisms of $\mathbb{C}$

Let’s assume that such a $\R$-automorphism $\varphi:\mathbb{C}\rightarrow \mathbb{C}$ exists. Then, since it is a field homomorphism, it must verify:
$$\varphi(a+bi)=\varphi(a)+\varphi(bi)=\varphi(a)+\varphi(b)\varphi(i)$$but because it is a $\R$-morphism, it fixes $\R$, so:
$$\varphi(a+bi)=a+b\varphi(i).$$So $\varphi$ would be entirely determined by $\varphi(i)$. But we have that $\varphi(-1)=\varphi(i^{2})=\varphi(i)^{2}$, which means that $\varphi(i)$ is a solution of the equation $X^{2}=-1$. This equation admits only two solutions, $i$ and $-i$.
We conclude that either $\varphi(a+bi)=a+bi$ (this is the identity map), or $\varphi(a+bi)=a-bi$, which is the complex conjugation (and it is indeed a $\R$-automorphism).