How many ways can we compactify $\mathbb{C}^n$?

Question

Below are two different ways we can compactify $\mathbb{C}$:

The first is "adding a point at infinity", the second is "adding a disc at infinity".

Intuitively, it looks like these are the only two ways we can compactify $\mathbb{C}$. (I declare that any other "disc" compactification is equivalent to the one I've drawn - although I'm not sure what the appropriate type of "equivalence" is.) Are there other non-trivial ways to compactify $\mathbb{C}$? Maybe there are other compactifications if we don't require it to be conformal?

Otherwise, are there nontrivial ways to compactify $\mathbb{C}^n$?

Answer 1

Every open connected $k$-dimensional manifold $M$ contains an open and dense subset homeomorphic to $R^k$. See my answer here.

Hence, every compact connected $2n$-dimensional manifold is homeomorphic to a compactification of ${\mathbb C}^n$.

Answer 2

Given a space $C$ a compactification of $C$ is a compact space $X$ that contains a homeomorphic copy of $C$ as a dense subset. A space can have many different compactifications, e.g. we have $S^1$ as one-point compactification and $[{-1},1]$ as two-point compactification of ${\mathbb R}$, whereby we of course write ${\mathbb R}\cup\{\infty\}$, resp., ${\mathbb R}\cup\{-\infty,\infty\}$ for these concepts.

The following is an example where we compactify ${\mathbb C}$ by adding a circle: The map $$f:\quad{\mathbb C}\to D, \qquad z\mapsto{z\over 1+|z|}$$ is a homeomorphism of ${\mathbb C}$ with the unit disc $D\subset{\mathbb C}$. It follows that $\bar D$ can be considered as a compactification of ${\mathbb C}$.