Quaternions are left handed, which is annoying but not a big deal by itself, but most importantly collapse the scalar with the pseudoscalar. For regular applications of unit quaternions in 3D euclidean space such as rotations, screw displacements and dual quaternions, that is not much of an issue since the scalar part is initially zero anyways (e.g. translation and rotations), it wholly contains the quadrature information after the operation (e.g rotation), or the scalar part is always separate from the vector quaternion (e.g. moments).

However if you want to see the quaternions as a vector subspace in problems of higher dimensions or in non-euclidean manifolds, for example path interpolation, finding integral solutions or representations, they fall short and it makes sense to use a Clifford algebra that is compatible with differential operations (e.g Lie) and gives consistent calculations across all dimensions.