In this paper the zeroes of the polynomial F(x, z) = 2x 4 −z 3 in Gaussian integers Z[i] are determined, a problem equivalent to finding the solutions of the Diophatine equation x 4 + y 4 = z 3 in Z[i], with a focus on the case x = y. We start by using an analytical method that examines the real and imaginary parts of the equation F(x, z) = 0. This analysis sheds light on the general algebraic behavior of the polynomial F(x, z) itself and its zeroes. This in turn allows us a deeper understanding of the different cases and conditions that give rise to trivial and non-trivial solutions to F(x, z) = 0, and those that lead to inconsistencies. This paper concludes with a general formulation of the solutions to F(x, z) = 0 in Gaussian integers. Results obtained in this work show the existence of infinitely many non-trivial zeroes for F(x, z) = 2x 4 −z 3 under the general form x = (1 + i)η 3 and c = −2η 4 for η ∈ Z[i].