The classical time scales calculus used in applied mathematics was originated as the result of investigating calculus of variations on time scales. It was introduced by its originator Stefan Hilger in the late eighties as an attempt to unify continuous and discrete problems in one theory. The main purpose of this paper is to introduce the Hilger quaternion numbers and deduce some of their fundamental properties. We extend the definitions of Hilger pure imaginary number, Hilger real axis, the Hilger alternating axis, and the Hilger imaginary circle within this context. We further introduce a generalized cylinder transformation to give a definition of the quaternionic exponential function on time scales. To the best of our knowledge, this does not appear to have been done in literature before.

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