Octonions and so on up are indeed a thing, but I don't think they do what you want. Even aside from the fact that they're restricted to power-of-2 dimensions,

Octonions and so on up are indeed a thing, but I don't think they do what you want. Even aside from the fact that they're restricted to power-of-2 dimensions, their algebraic properties get worse as you iterate the Cayley-Dickson process. The octonions aren't even an associative algebra, although they do have some weaker associativity properties which I'll skip detailing here. The quaternions are as far as most mathematicians are willing to go -- non-commutativity is commonplace, but who wants to deal with non-associativity? But while the octonions at least have some mathematical relevance (they're actually connected to various exceptional objects, such as the exception Lie group G_2!), the sedenions and beyond basically don't. They have a tiny bit of associativity but not enough that they connect to any things or that hardly anyone wants to study them -- and worse yet, there are zero divisors so cancellation (ab=ac => b=c for nonzero a) doesn't even hold. (Inverses exist, yes, but without associativity, inverses don't imply cancellation! And therefore aren't much use.) As another commenter mentioned, what you might be looking for instead if it's orthogonality you're focused on is Clifford algebras (aka geometric algebra). However, if you want to get the complex numbers or quaternions out of it, you'd need to use a negative-definite quadratic form -- if you use a positive-definite one, you'd instead get the split-complex numbers, which are much less interesting (and you'd get something similar instead of the quaternions).