Interesting links with special relativity and quantum theory but these fractals seem to already exist under many names such as Indra's pearls, IFS, pseudo-kleinians.
IFS is a generic name for "iterated function system". Indra pearls are different. They live on the Poincare disc (non-compact space), not on the sphere that is compact. Etc. etc. One may say: all fractals are similar. And yet there are many different families of fractals with different properties. The devil is in the details.
For instance, according to a recent paper by Andrew Vince et al. on Mobius IFS, the last fractal ("para") in the Generator has no rights to exist. Vince proved (with Barnsley) that parabolic transformations can not generate fractals. And yet we have something that looks like a fractal and walks like a fractal. It is no yet clear if it quacks like a fractal. Vince himself was puzzled when saw this fractal
Since most fractal generators work on a repeated set of mobius transformations, since these are the conformal set (which also exist in more dimensions) required for the result to be fractal and not a stretched out shape.
Those on the sphere are partially stretched. They are not made of contractions.
Is there a suggestion that some real world quantum wave functions would be fractal like these?
It is not that simple. The distribution (measure) on the sphere represents a density matrix (mixed state), not a wave function. Moreover, because of linearity of quantum mechanics many different distributions represent the same mixed state. So, according to the orthodox interpretations of quantum theory these fractal measures will not be observable. Yet there are also non-orthodox interpretations (you have to get rid of the orthodox interpretation of the Heisenberg's uncertainty relations first) that suggest that these fractal shapes can leave experimental traces. So there may be some new adventure ahead...
P.S. And indeed, there are fractals made of Mobius transformations living in higher dimensions. Especially nice are those in 4 dimensions, since in 4D we have beautiful Platonic solids (24 cell and 600 cell). They will be added to the generator (3D slices) in due time.