My question is whether $\mathbb{R}^*\oplus \mathbb{R}^* \simeq \mathbb{C}^*$. I think this isn't true because $(0,1)^2 = (0,1)$, but $i^2 = -1$. I can't quite figure out how to make this rigorous though.
I do know that $\mathbb{R}\oplus \mathbb{R} \simeq \mathbb{C}$.
You're on to something, but you don't know in advance whether or not your potential isomorphism will send $(0,1)$ to $i$. Edit: And as pointed out by Thomas Andrews, $(0,1)$ is not an element of $\mathbb{R}^* \oplus \mathbb{R}^*$.
How many elements in $\mathbb{R}^* \oplus \mathbb{R}^*$ have their square equal to $-1 = (-1,-1)$? How many elements in $\mathbb{C}^*$ have their square equal to $-1$?
You could also ask how many elements have their square equal to $1$.