Alternative Compactification of $\mathbb{C}$

Question

Of course, there is the compactification of $\mathbb{C}$, $\mathbb{C}\cup\{\infty\}$, which allows us to correspond the complex plane to the unit sphere. Is there any use in compactifying $\mathbb{C}$ with something like $\mathbb{C}\cup\{\theta\infty|\theta\in[0,2\pi)\}$ (pardon my abuse of notation) which essentially adds a point at infinity for EACH direction around the plane?