  Over the last years, considerable attention has been paid to the role of the
quaternion differential equations (QDEs) which extend the ordinary differential
equations. The theory of QDEs was recently well established and it has wide
applications in physics and life science. This paper establishes a systematic
frame work for the theory of linear quaternion dynamic equations on time scales
(QDETS), which can be applied to wave phenomena modeling, fluid dynamics and
filter design. The algebraic structure of the solutions to the QDETS is
actually a left- or right- module, not a linear vector space. On the
non-commutativity of the quaternion algebra, many concepts and properties of
the classical dynamic equations on time scales (DETS) can not be applied. They
should be redefined accordingly. Using $q$-determinant, a novel definition of
Wronskian is introduced under the framework of quaternions which is different
from the standard one in DETS. Liouville's formula for QDETS is also analyzed.
Upon these, the solutions to the linear QDETS are established. The Putzer's
algorithms to evaluate the fundamental solution matrix for homogeneous QDETS
are presented. Furthermore, the variation of constants formula to solve the
nonhomogeneous QDETs is given. Some concrete examples are provided to
illustrate the feasibility of the proposed algorithms.

Classical Analysis and ODEs