  We consider the competition between decoherence processes and an iterated
quantum purification protocol. We show that this competition can be modelled by
a nonlinear map onto the quaternion space. This nonlinear map has complicated
behaviours, inducing a fractal border between the area of the quantum states
dominated by the effects of the purification and the area of the quantum states
dominated by the effects of the decoherence. The states on the border are
unstable. The embedding in a 3D space of this border is like a quaternionic
Julia set or a Mandelbulb with a fractal inner structure.
