  In this paper, we introduce a Gauss-Newton method for solving the complex
phase retrieval problem. In contrast to the real-valued setting, the
Gauss-Newton matrix for complex-valued signals is rank-deficient and, thus,
non-invertible. To address this, we utilize a Gauss-Newton step that moves
orthogonally to certain trivial directions. We establish that this modified
Gauss-Newton step has a closed-form solution, which corresponds precisely to
the minimal-norm solution of the associated least squares problem.
Additionally, using the leave-one-out technique, we demonstrate that $m\ge O(
n\log^3 n)$ independent complex Gaussian random measurements ensures that the
entire trajectory of the Gauss-Newton iterations remains confined within a
specific region of incoherence and contraction with high probability. This
finding allows us to establish the asymptotic quadratic convergence rate of the
Gauss-Newton method without the need of sample splitting.
