I want to prove that the mapping of the complex plane to the complex plane is a closed map.
Only way I can think of is to prove first that it is a proper map.
Is there a more straightforward way ?
It's obvious that is closed, and since it leaves fixed it behaves nice with respect to unbounded closed sets (an unbounded set is closed in the plane iff the set plus is closed in the sphere). ( denotes the Riemann sphere)
Not at all: Take where the natural projection on the real axis, then fixes (in the sense that there is a mapping of the sphere to the sphere that coincides with via stereographic projection, ie. as usual), but it's not difficult to see that is not closed. On the other hand if is a constant, we can extend to the sphere and this is, trivially, a closed continous map that doesn't fix .
What I meant to say was that obviously your mapping takes compact sets to compact sets (both in the plane), so we only need to check whether it sends unbounded closed sets (say ) to closed sets ( say), but by adding to we get a closed set in the sphere and thus is closed in the sphere, but by the first post is closed in the plane, but this is exactly and we're done.