Imaginary Scators Bound Set Under the Iterated Quadratic Mapping in 1 + 2 Dimensional Parameter Space

The quadratic iteration is mapped within a non distributive imaginary scator algebra in 1+2 dimensions. The Mandelbrot set is identically reproduced at two perpendicular planes where only the scalar and one of the hypercomplex scator director components is present. However, the bound three dimensional S set projections change dramatically even for very small departures from zero of the second hypercomplex plane. The S set exhibits a rich fractal like boundary in three dimensions. Periodic points with period m, are shown to be necessarily surrounded by points that produce a divergent magnitude after m iterations. The scator set comprises square nilpotent elements that ineluctably belong to the bound set. Points that are square nilpotent on the mth iteration, have preperiod 1 and period m. Two dimensional plots are presented to show some of the main features of the set. A three dimensional rendering reveals the highly complex structure of its boundary.