This paper is focused to find the potential solutions of a binomial Fermat type functional equation in \({\mathbb {C}}^2\) where the equation involves operators formed by linear combinations of functions, shifts, and derivatives as it allows for a more comprehensive understanding of the underlying structures and relationships. Our approach is the first to unify the treatment of the function and its two important variants within a common framework to analyze the impact of the operator on the solutions of the Fermat type functional equation in two dimensional complex field. In our another attempt, for a specific subclass of the combination of the operators, we have been able to extend our result for n dimensional complex plane.
