[gugagore]

I think the answer to that question is that it doesn't "stop", but I'll try to offer a reason that isn't "octonions exist", but instead goes in a different direction.

Geometric Algebra can capture the structure of both complex numbers and quaternions, and also the structure of the dot products, cross products, and the different kinds of vectors that arise from those operations.

To be clear, matrix multiplication can also capture the structure of complex numbers [1] and quaternions [2]. There might also be a concise reference to matrix representations of some geometric algebras, but I didn't find one. So matrices are kind of one way to not "stop at 3D", but the structure is almost too uniform (which on one hand makes it too general, and on the other hand makes it not general enough), I'd say). Sure, with a matrix you can represent rotations in 4D, but you still need to operate on vectors only. Geometric algebra, if it does have a matrix representation, gives names to special kinds of matrices and special kinds of vectors.

[1] https://en.wikipedia.org/wiki/Complex_number#Matrix_represen... and [2] https://en.wikipedia.org/wiki/Quaternion#Matrix_representati...