Isn't this just a complex plane with weightings attached?
Oh, silly me. It's the complex plane with an extra point for infinity. Silly me. (This point is, of course, represented as a line around the circumference of the plane. No, you can't have a circumference around something infinite, but it makes it easier to draw. Anyway, then you can contort the plane into an 'almost sphere' with this final point making it a real sphere - the RiemannSphere.
Whahey! I remembered something!
(Unless I remembered wrong, and the extended complex plane is really the complex plane with a line cutting out the zero point - in a display of ironic naming practices.)
Hi all. Consider the extended complex plane as `the complex plane union the point at infinity'. This implies that the mapping z -> 1/z maps the extended plane to the extended plane - the points at zero and infinity are interchanged. However, on a normal complex plane, this would not be the case.
The extended plane is a useful thing to consider for nice topological reasons and analytical reasons, eg the RiemannSphere.