No, it's not an alternative. You're conflating the grab bag of topics in an undergraduate linear algebra class with the subject of linear algebra.

Linear algebra is the study of linear operators on vector spaces over fields (a special case of modules over rings). Some vector spaces are inner product spaces, but most are not.

Exterior algebra is an example of multilinear algebra. Clifford (or geometric) algebras are constructed as an algebra over a vector space. Quaternions come via a different route: instead of constructing a multilinear algebra, they are one of a handful of very special algebras like C and the octonions.