Would it be judicious to define the Hilbert space of Quantum Mechanics over the quaternion numbers instead of the complex numbers? We can define a tensor product by the Hamilton numbers $\bf Q$:

$H_{\bf Q}={\bf Q} \otimes_{\bf R} H$

with products:

$q(q' \otimes \psi)=(qq')\otimes \psi$

$(q \otimes \psi)q'=(qq')\otimes \psi$

with $H$ the complex Hilbert space. So that we have the brackets $<q \psi|\phi>=\bar q<\psi|\phi>$, $<\psi q |\phi >=<\psi | \bar q \phi>$ and $<\psi |\phi q>=<\psi |\phi> q$.

The observables would be right linear operators: $A|\psi q>=A|\psi >q$.