How come the author makes the assertion that $$(\alpha(i))^2=\alpha(i^2)?$$
I understand that $j=\alpha(i)$ must be imaginary, since it can't map to a real number or else the $\mathbb R$-automorphism won't be injective, hence, well, no longer an automorphism.
But why can't perhaps $\alpha(i)=2i$?
I can't tell why $\alpha(x+yi)=x+2yi$ isn't considered an automorphism given how it's a bijection $\mathbb C \to \mathbb C$ and we still have $$(\forall k \in \mathbb R): \quad \alpha(k)=k.$$
A field automorphism $f\colon \mathbb{C}\rightarrow \mathbb{C}$ not only satisfies $f(x+y)=f(x)+f(y)$, but also $f(xy)=f(x)f(y)$. For $x=y=i$ we obtain $(f(i))^2=f(i^2)=f(-1)=-1$. The last equality follows because $f$ fixes real numbers, since it is an $\mathbb{R}$-automorphism of $\mathbb{C}$.