For molecules with complex and competing magnetic interactions, it is often the case that the lowest energy Hartree-Fock solution may only be obtained by removing the spin and time-reversal symmetry constraints of the exact non-relativistic Hamiltonian. To do so results in the complex generalized Hartree-Fock (GHF) method. However, with the loss of variational constraints comes the greater possibility of converging to higher energy minima. Here, we report the implementation of stability test of the complex GHF equations, along with an orbital update scheme should an instability be found. We apply the methodology to finding the local minima of several spin-frustrated hydrogen rings, as well as the non-collinear molecular magnet Cr3, illustrating the utility of the broken symmetry GHF method and some of its lesser-known nuances.