  M\"obius transformations of the extended complex plane are at the crossroads
of many interesting topics, e.g., they form a group under composition, are the
simplest form of rational function, and are a path to Lie theory. Quaternionic
transformations are a subgroup of M\"obius transformations isomorphic to
rotations of the Riemann sphere, which also represent quaternion conjugation.
These representations yield formulas for the axis and radians of rotation, and
thereby portray each particular quaternionic transformation as part of a
continuous orbit of rotations sharing a common axis.
