[math-fun] Complex projective spaces

Can someone give a geometrical or topological description of the shape of the complex projective space CP^n ? While CP^1 is the 2-sphere, some searching reveals a few results. CP^n is simply connected (in contrast to RP^n). CP^n for n>1 is not homeomorphic to the sphere S^2n. The quotient of CP^2 by complex conjugation is S^4. I'm writing a paper on the optics of polarized light, and I need to understand CP^2 and CP^3. -- Gene