In this paper, we show that the Stokes-Mueller formalism can be reformulated in terms of quaternions and that the quaternion algebra is a suitable alternative presentation of the formalism of Mueller-Jones states that we have recently described [J. Opt. Soc. Am. A34, 80 (2017)JOAOD60740-323210.1364/JOSAA.34.000080]. The vector and matrix states associated with the Mueller matrices of nondepolarizing optical systems are different representations that are isomorphic to the same quaternion state, and this quaternion state turns out to be the rotator of the Stokes quaternion. In this work, we study the properties of this general quaternion state and its application to the calculus of polarization effects. We also show that the coherent linear combination of nondepolarizing optical media states and depolarization phenomena can be reformulated in terms of quaternion states.