You don't even need the complex part for this. You can do the infinity-projection trick on the real numbers alone as well: https://en.wikipedia.org/wiki/Projectively_extended_real_lin...

A similar trick (point at infinity or ideal point) is used in projective geometry to distinguish between directions (vectors) and places (points) by using coordinates only: https://en.wikipedia.org/wiki/Projective_geometry

But if you actually want to do calculations with infinities and infinitesimals the surreal numbers might be better suited for that: https://en.wikipedia.org/wiki/Surreal_number