Is it possible to get an holomorphic immersion form $\mathbb{C}$ to $\hat{\mathbb{C}}$ which is surjective?
Here immersion means that $f'(z)\neq 0$ when $f(z)$ is finite and ${\left(\frac 1f\right)}'(z)\not=0$ when $f(z)=\infty$. Of course if you show that $f$ is a covering you get a contradiction, but don't succeed in proving that.