geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
The 2-sphere with its canonical structure of a complex manifold is called the Riemann sphere.
As such this is complex projective space $\mathbb{C}P^1$.
The biholomorphisms, i.e. the bijective conformal transformations from the Riemann sphere to itself are the Möbius transformations.