For example, the rotation plus translation of a point using the language of quaternions is written as
$Q(0,x,y,z)Q^* + T$
where $Q$ is the unit quaternion, $(x,y,z)$ is the point, and $T$ is some translation vector
This looks remarkably similar to computing the expected state in quantum mechanics, namely
$\langle Q^*|V|Q^*\rangle$ where $Q^*$ is some quantum state and $V$ is some operator
Can an analogy be made between the quarternion and quantum mechanics in terms of the bra-ket formulation?