go for the Reimann sphere
The Riemann sphere extends the complex plane by adding one point at Infinity. It is the plane plus one point, say it the North pole, with some good math formulas to ensure nice behaviuors, in particular to avoid functions discontinuity and other goodies like Conformal map i.e. circles are mapped into circles.
I dedicate this work to my father Umberto Casati and the great mathematician Luigi Bianchi who was a great inspiration for me.