Dear Jonathan,
thanks for the very interesting essay (with a high vote from my side). Your consideration of octonions keep me interested
again in this interesting topic. I considered the hypercomplex numbers years ago but forgot
it.
Some remarks from topology. There is a topological view on these normed algebras (C, H, O).
The unit elements are spheres: S^1, S^3 and S^7. Now there is a "topological" proof that
C,H,O are the only normed algebras. If there is a non-zero vector field on the sphere S^1,
S^3, S^7 then there is a non-zero element of the algebra which can be reversed (allowing
division). But one needs more: every element of the algebra (except zero of course) must be of this form. But then you need as much as non-zero independent vector fields like the dimension of the sphere. A deep theorem states now that only the S^1, S^3 and S^7 have this property so that the only normed algebras are C, H and O. (BTW I used this result for a no-go theorem in quantum computing, see my paper.)
Nonassociated algebras are not so special in math. A large class are the Lie algebras (with the Jacobi identity instead of the associative law). But octonions are more than that. This algebra is deeply connected to the exceptional Lie groups (like E_8, E_7 or G_2). In geometry there are special manifolds with a calibration (a linear form with special properties). There, the octonions appear as G_2 structures on 7-dimensional manifolds. M theory as part of string theory used them.
But you rather ask for low-dimensional (3D, 4D) counterparts. I'm specialized in this dimensions and here it is: a very large class of 3-dimensional manifolds (arithmetic hyperbolic 3-manifolds) is constructed by using the quaternion algebra and its properties are strongly related to (or determined by) this algbera.
4-dimensional manifolds (simply-connected i.e. where all closed curves can be contracted to a point) are classified by the intersection form: a bilinear form with the intersection number of the embedded surfaces. This bilinear form is taken over the integers. In the classification of these forms, the so-called E_8 lattice played a fundamental role. But this lattice is nothing else then the integer octonions (Kirmse integers). So, your octonions are directly related to the spacetime (seen as 4D manifold) and you don't need any extra dimensions.
If you like please have a look into my essay where I used topological methods to understand our brain.
All the best
Torsten