In this work, we introduce a new form of the quaternionic fractional uncertainty relation within the framework of quaternionic quantum mechanics. This is closely associated with the Li-Ostoja-Starzewski fractional gradient operator, characterized by an order range of 0 ≤1. We explore a novel Quaternionic Schrödinger equation and its specific implications, particularly addressing solutions that lead to the emergence of position-dependent mass. Additionally, we validate the theory by comparing it against the observed maximum wavelengths in the 1,3,5-hexatriene molecule.

The introduction of complex numbers marked a significant leap in mathematics, introducing the imaginary unit i to represent the square root of -1. This innovative concept proved invaluable in solving equations involving square roots of negative numbers. The extension to quaternions involved introducing additional imaginary units denoted as j and k. A quaternion in an interesting concept that extends complex numbers to 4-D.The manuscript is about using quaternions to calculate wave packets in one-dimension, and anti-hermitian operators to obtain the results in quaternionic form including expectation values of position, momentum and energy. The results are compared to the exisiting results on complex wave packets in one-dimension.