Quote by bapowell
Clifford algebras are particularly necessary for understanding spinor representations and spin groups. Depending on what you're interested in studying, they could be important (a good knowledge of the mathematical underpinnings of SM and QFT will require knowledge of Clifford algebras). I'd say that quaternions (and octonions) are less important: quaternions offer an alternative to the tensor formulation of GR, and are not something I'd concern myself with on an initial read through.
I would certainly get up to speed on your linear algebra and group theory (in particular Lie groups and Lie algebras which are widespread in particle physics) before I'd worry about Clifford algebras and quaternions.

The problem is that I seem to have been able to justify the use of quaternions and octonions from purely logical reasons. And now I wonder if this is a road into physics. So I feel like I need to go from these hypercomplex numbers to SM physics. You seem to indicate that you can go from the SM to hypercomplex numbers. But can one go from hypercomplex numbers to the SM?