  Positive matrices in SL(2,C) have a double physical interpretation; they can
be either considered as "fuzzy projections" of a spin 1/2 quantum system, or as
Lorentz boosts. In the present paper, concentrating on this second
interpretation, we follow the clues given by Pertti Lounesto and, using the
classical Clifford algebraic methods, interpret them as conformal maps of the
"heavenly sphere" S^2. The fuzziness parameter of the first interpretation
becomes the "boost velocity" in the second one. We discuss simple iterative
function systems of such maps, and show that they lead to self--similar fractal
patterns on S^2. The final section of this paper is devoted to an informal
discussion of the relations between these concepts and the problems in the
foundations of quantum theory, where the interplay between different kinds of
algebras and maps may enable us to describe not only the continuous evolution
of wave functions, but also quantum jumps and "events" that accompany these
jumps. Paper dedicated to the memory of Pertti Lounesto.
