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In this work, we introduce a fractional generalization of the classical Moisil-Teodorescu operator that provides a concise notation for presenting a mathematical formulation of physical systems in fractional space from various branches of science and engineering. The method used in this article, called the Stillinger’s formalism, is combined in a novel way with quaternionic analysis. As a concrete application, a quaternionic reformulation of a fractional time-harmonic Maxwell system is established, thus showing a deep relation between it solutions with those of the perturbed fractional Moisil-Teodorescu operator. Furthermore, the fractional constructions here will find further applications in several applied research areas such as in hydrodynamics and magneto-hydrodynamics.