Mueller Quaternion with Matrix Coefficients for Depolarizing and Nondepolarizing Media

It is shown that a Mueller quaternion can be formulated as a quaternion with matrix coefficients, and this Mueller quaternion  transforms the Stokes quaternion for depolarizing  process as well as for nondepolarizing process. Mueller quaternion keeps track of the phase acquired by the Stokes quaternion and, in general,  quaternion states of optical media comprises all properties of Jones, Mueller-Jones and Mueller matrices.