  In this work, we point out that the Q/U Stokes parameters and E/B mode
polarizations are the four components of a unique quaternion, which describes
at the same time the directions and the parity states of spherical linear
polarizations. We then point out that, with this polarization quaternion, the
mathematical form of all Q/U and E/B transforms are greatly simplified, to an
extent that requires only one quaternion multiplication for each transform. A
preliminary application of the polarization quaternion is shown as an example
to detect peculiar pixel domain patterns within the E- and B-families, which
are the former and latter halves of the polarization quaternion.
