Hi Colin,
Thanks for an intriguing perspective.
> Quaternions arise from the solution of an elementary two-dimensional polynomial, and can be visualized in terms of amplitude and phase spectra by means of a multidimensional Fourier transform.
I share your enthusiasm for quaternions and I advocate the use of Geometric Algebra. In my essay Software Cosmos you can see how these play into both the explicate world that we measure and also an implicate space. These two spaces (one observable, one not) might be linked via just such a Fourier transform.
> Two forms correspond to slices through the octonion vector space and are essentially two-dimensional. The third form is a tetrahedron in three dimensions.
> The Fano plane shows the seven imaginary elements of an octonion having seven sets of quaternion cyclic ordering given by three sides of the triangle (426, 635, 514), the three altitudes (473, 671, 572), and the circle of midpoints (123).
Sounds like this would be useful in constructing models for quantum contextuality. Michel Planat discusses such models in his essay, including the Fano plane. My comment to Michel suggested he look at k-rational points based on Q(phi), the extension of the rationals by the golden ratio. The coordinates of a tetrahedron (and many other polytopes) would qualify.
> I can imagine the picture of existence that emerges could be analogous to a quantum computer operating coherently in the Higgs condensate.
I hope you get a chance to check my essay, as I think my picture nicely complements your views.
Hugh