Projective lines which can be viewed in some sense as surfaces

Question

The complex projective line can be viewed as the $2$-sphere.

I'd appreciate some examples of other projective lines (over any field or even ring) that can be viewed in some sense as a surface. I am purposefully being vague when I say "in some sense" because I do not have any precise way I would like to view the projective lines as just yet.

It could be that that the projective line is topologically equivalent to some surface, or it could just be bijective to a particular surface, or it could have any other abstract or concrete interpretation as a surface.

Answer 1

Your example generalizes to any degree 2 field extension $E/F$: being a two-dimensional $F$-vector space, $E$ can be viewed as a plane over $F$, and the projective line over $E$ can be viewed as an inversive plane over $F$.