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Quaternionic calculus is significant in applied mathematics. It is closely referred to mathematical physics and engineering sciences. Our aim is to present some advances in the investigations of the quaternion dynamic equations on time scales and the Henstock–Kurzweil ∆-integral (abbreviated as HK-∆-integral) and to introduce the corresponding quaternionic version of the HK-∆-integral (abbreviated as HKQ-∆-integral). Some basic properties of the HKQ-∆-integral are demonstrated, which would be helpful in the future research in this direction. [ABSTRACT FROM AUTHOR]