I've noticed that the pictures illustrating the effect of Möbius transformations on the Riemann sphere (after stereographic projection to the plane) resemble the phase portrait of a vector field. For instance:
This is the representation of a hyperbolic transformation (from Wikipedia), and seems very similar to the phase portrait of a field generated by a positive and a negative charge located at the fixed points of the transformation (that is, of an irrotational and incompressible field). One could ever think of linking the fact that poles have integer order to the discretization of charges, using Gauss's divergence theorem. I guess the analogy arises from conformal mappings satisfying Laplace's equation, but I wonder how far can one push it.
This is where I'm doubting: in the example cited above, hyperbolic transformations are the stereographic projection of discrete rotations of the sphere. In general, Möbius transformations are discrete mappings from the Riemann sphere to itself, whereas a vector field suggests a continuous flow. Could we rigorously obtain a one-parameter subgroup of continuous transformations from those infinitesimal, discrete transformations, and could we define a (unique) vector field corresponding to this continuous subgroup (for example, writing each transformation as a differential equation)? How?
Please excuse me if my approach is hand-waving, I don't know much about this subject.