I can't tell whether this has much substance. It doesn't seem to ease calculations in a classical simulation of a quantum system. Also not sure whether it adds any substantial understanding beyond the Bloch sphere view of a qubit, especially when you start extending to multi-qubit systems. Given the normalisation condition for a qubit amplitude, it is natural to define a qubit using three angles for which a quaternion is well suited ... And which is already captured by the Bloch sphere picture.
Another thing is that Pauli matrices which are unitary transforms of qubits represented in the conventional column vector form are themselves quaternion bases. So in essence what this is saying is that you can cover the space of qubits using exponents of Pauli matrices applied to eigenstates.
Don't think so? Happy to hear alternative views.