Is my proof correct? (Isomorphism)

We must show that $\displaystyle \mathbb{R}$ and $\displaystyle \mathbb{C}$ are NOT isomorphic. Consider the mapping $\displaystyle \alpha : \mathbb{C} \rightarrow \mathbb{R}$. Then we have $\displaystyle \alpha(i)=a$ which can be written as $\displaystyle \alpha(i)^2=a^2$. So, $\displaystyle \alpha(i^2)=a^2$ which is the same as $\displaystyle \alpha(-1)=a^2$. We can split this apart to get $\displaystyle \alpha(-1)+\alpha(0)=a^2$. This shows that $\displaystyle -1=a^2$. But nothing in $\displaystyle \mathbb{R}$ squares to $\displaystyle -1 $. Therefore they are not isomophic.