Let $S$ be the Stone-Cech compactification of $\mathbb C$. Then any meromorphic function $f:\mathbb C \to \mathbb{CP}^1$ extends to $S$.

Do the meromorphic functions separate the points of $S$?

I doubt there are enough such functions, though I could also believe that this could depend on our set theory.

If not, "what is" the space $S/\!\sim$, where $p\sim q$ if $f(p)=f(q)$ for all meromorphic $f$?

Again, I could easily believe that this question has very different answers depending on one's set theory.

I don't have any especially good reason for asking this question. Feel free to retag.