>>2
Sounds vaguely like the Riemann Sphere coordinate mapping:
http://en.wikipedia.org/wiki/Riemann_sphere
But you can approach infinity when working on the Riemann Sphere, and it's a 2-manifold in 3-space, rather than your line (plus a point) in 2-space. You can even construct a legitimate function which gives you the euclidean distance between the mapping of a Complex point and infinity. Not that this would give us any particularly useful information, but it is a method you could use to quantify "close to infinity."
Also of interest: "[It] makes expressions such as 1/0=∞ well-behaved and useful."