I'd be interested to know your opinion on space-time algebras. They seem like they would provide a nice way of unifying spatial rotations and Lorentz boosts, but that may be blind optimism as your last sentence seems to have been written to describe me specifically...
Spacetime algebra is just the Clifford algebra Cl_1_3(R), so all of the above applies.
> They seem like they would provide a nice way of unifying spatial rotations and Lorentz boosts
Yes and no: the right way to unify rotations and boosts is to consider them as the orientation-preserving elements of the Lorentz group (sometimes called the proper Lorentz group). You can construct this from the corresponding Clifford algebra, but it's somewhat technical and not physically well-motivated until you start dealing with spinors. It's also the group of symmetries of spacetime that leave the origin unchanged, which is far, far more natural.
