Find a homomorphism $\phi: \Bbb C \to \Bbb R^*$ such that $\phi(i) = 2$.
I can't think of anything here, I can only get $\phi(a+bi) = 2^b,$ but I'm pretty sure I can't just extract the imaginary part like that.
If it was $\mathbb C^*$ instead of $\mathbb C$ then I know I can use absolute value / magnitude somehow but in this case it wouldn't be a homomorphism.
You can.
Let $x=a+ib,y=c+id\in\Bbb C$. Then
$$\begin{align} \phi(x+y)&=\phi((a+c)+i(b+d))\\ &=2^{b+d}\\ &=2^b2^d\\ &=\phi(a+ib)\phi(c+id)\\ &=\phi(x)\phi(y). \end{align}$$
Thus $\phi:\Bbb C\to \Bbb R^*$ is indeed a homomorphism.
Moreover, $\phi(i)=\phi(0+i1)=2^1=2$.