The division by zero shows the strong discontinuity at the point at infinity, however, the Riemann sphere model and stereographic projection mapping are fundamental and beautiful.
Many people feel strange feelings for the strong discontinuity; however, the strong discontinuity appears in the universe naturally as we see from our new and many concrete results since Euclid.
However, the beautiful Puha's horn torus model may be accepted with great pleasures as our space idea. In particular, note that the domains $\{|z|<1\}$ and $\{|z|>1\}$ are completely conformally equivalent and so the completely symmetric property of the corresponding domains on the horn torus is very fine and from this viewpoint, the Riemann sphere model will be curious, in particular, at the point at infinity and the point at infinity will be vague.