  Quaternionic analysis relies heavily on results on functions defined on
domains in $\mathbb R^4$ (or $\mathbb R^3$) with values in $\mathbb H$. This
theory is centered around the concept of $\psi-$hyperholomorphic functions
i.e., null-solutions of the $\psi-$Fueter operator related to a so-called
structural set $\psi$ of $\mathbb H^4$. Fractional calculus, involving
derivatives-integrals of arbitrary real or complex order, is the natural
generalization of the classical calculus, which in the latter years became a
well-suited tool by many researchers working in several branches of science and
engineering. In theoretical setting, associated with a fractional $\psi-$Fueter
operator that depends on an additional vector of complex parameters with
fractional real parts, this paper establishes a fractional analogue of
Borel-Pompeiu formula as a first step to develop a fractional
$\psi-$hyperholomorphic function theory and the related operator calculus.
