It is harder! Calabi's trick is one attempt at making it easier. Hmmmmm. If you're ok with the real version... well another way to write down the real version

It is harder! Calabi's trick is one attempt at making it easier. Hmmmmm. If you're ok with the real version... well another way to write down the real version is "lines through the origin in real 3-space" (that's what the equivalence relation turns out to be). How do you feel about "lines through the origin in real n-space"? How about "planes through the origin in real n-space"? If we start with "complex lines through the origin in complex 3-space" (the complex projective plane we're trying to understand), and throw away all the complex stuff, then that's "some real 2-planes through the origin in real 6-space", i.e. it's a subset of planes through the origin in real 6-space. It's a proper subset because every complex line is a 2-plane but not every 2-plane in 6-space is a complex line. So maybe that's a start? It's not a perfect way to visualize it because (a) 2-planes in 6-space aren't exactly easy to visualize either, and (b) okay but which planes in 6-space are the special ones that come from the complex structure that we ignored earlier. But maybe it's something.