  Quaternionic analysis offers a function theory focused on the concept of
$\psi-$hyperholomorphic functions defined as null solutions of the
$\psi-$Fueter operator, where $\psi$ is an arbitrary orthogonal base (called
structural set) of $\mathbb H^4$.
  The main goal of the present paper is to extend the results given in
\cite{BG2}, where a fractional $\psi-$hyperholomorphic function theory was
developed. We introduce a quaternionic perturbed fractional $\psi-$Fueter
operator calculus, where Stokes and Borel-Pompeiu formulas in this perturbed
fractional $\psi-$Fueter setting are presented.
