Consider the upper half plane $\mathbb{H} = \{ z \in \mathbb{C} : \text{im}(z) > 0 \}$ as a Riemann surface.
I suspect that the compactification of $\mathbb{H}$ is the Riemann sphere $\mathbb{P}_{\mathbb{C}}^1$. Or maybe it's just a point $*$?
How would I find the compactification of $\mathbb{H}$?