So far, I have said that if $U$ is proper, open and simply connected, then its complement contains an open set. Suppose $f$ is a conformal equivalence form $\mathbb{C}$ tp $U$. Consider the induced meromorphic function on the Riemann sphere. By Casorati-Weierstrass, since the image does not include the set missed in $\mathbb{C}-U$, so the singularity at infinity is a pole. Since $f$ is bijective and analytic, then by the Valency Theorem the pole must be simple.
I don't know where to go from here.