Let \(m\), \(r\) be positive integers with \(r\geq 2\) and \(m\) even. Define integers \(A\) and \(B\) by \(A+B\sqrt{-1}=(m+\sqrt{-1})^r\). The author proves that the equation \(| A| ^x+| B| ^y=(m^2+1)^z\) has unique solution \((x,y,z)\) in positive integers, provided that either \(r\equiv 4 \pmod 8\) or \(r\equiv 6 \pmod 8\) and \(m^2/\log (m^2+1)\geq r^3 / \log 2\).
The proof is based on a study of small divisors of the components of a solution. Using various results on generalized Fermat equations, sharp bounds for solutions are obtained. Most of the candidate solutions are thereafter excluded by relating them to rational points on suitable elliptic curves or by computing Jacobi symbols.