It's a nice extension of the plane to get some results. For example https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem (Gemini recommends me this video, but I didn't see it https://www.youtube.com/watch?v=5OYLnMyesX4&t ). The idea is that in the complex protective plane, two lines intersect in exactly a point, and two circles intersect in exactly 4 points (yes 4). It's very easy to count intersections there, so after you counted them there you can try to count how many of them are in the usual plane and how many are "fake" intersections in the extension.

There are many similar results, but this is one of the most easy to understand.