'complex manifolds' → Riemannian sphere, mapping on horn torus & more (click image or scroll down)
  ⇐ 
complex plane and Riemann sphere are bijective mappings between one another, only infinity has to be treated separately
Vyacheslav Puha (Вячеслав Пуха) describes an additional mapping as stereographic projection, using the horn torus,
to solve various problems with infinity, especially for treating the mathematically forbidden operation 'division by zero'
allowing an analytical alteration, the horn torus represents the set of complex numbers in a static conformal way too
 
in my (Wolfgang W. Daeumler's) horn torus model the complex numbers are represented in a totally different way:
the horn tori are dynamic! roughly speaking: real numbers correspond to the angle of rotation (number of rotations)
and imaginary numbers are represented by the length of the unrolling line, which the horn torus travels on the axis
when the bulge of the horn torus executes the revolution around itself and along (!) this centered (symmetry) axis
 

 
standard dynamic horn torus, version  00 ( spin + / time + )
click an arrow to change direction of turn:   ↔ ↕ ↕↔
(click animation for nested horn tori)
 
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