Quaternions are the even part of the Clifford (aka Geometric) algebra associated to a three-dimensional space, so the connection is quite close. This case is also quite special because the quaternions form a division algebra over the real numbers, meaning that each nonzero quaternion has an inverse. Finite-dimensional real division algebras are quite rare: indeed the only ones up to isomorphism are the real numbers, complex numbers, and quaternions.

Geometric algebra is great if you want to extend things to higher dimensions, or spaces with different metric signatures (the 4-dimensional spacetime for example). But quaternions should not be discounted just because they fit into a generalisation - they have many quite special properties all of their own.