Description
Neural quantum states (NQS) offer a powerful variational framework for many-body systems; however, extending them to non-Hermitian (NH) Hamiltonians raises fundamental questions regarding optimization, symmetry, and architectural bias. In this study, we introduce a complementary optimization framework for progressive adaptive state search (COMPASS) based on biorthogonal adaptive recurrent neural quantum states. Our approach combines an adaptive autoregressive architecture with a biorthogonal variational Monte Carlo scheme and a complementary optimization scheme that alternates adiabatically between energy and variance minimization. This enables stable convergence to ground-state eigenpairs while avoiding Markov chain sampling through an exact autoregressive generation. We demonstrate that for parity-time ($PT$)-symmetric Hamiltonians, unconstrained complex ansätze can spontaneously break $PT$ symmetry during optimization--even in the unbroken phase--leading to spurious imaginary energies. Real-valued ansätze in an appropriate basis naturally constrain the optimization to the correct physical manifold. Conversely, for generic NH Hamiltonians without symmetry protection, complex ansätze are essential for accurately capturing complex ground-state properties. Our results establish that physically informed ansatz selection is crucial for reliable NH simulations. By combining adaptive architectures, biorthogonal optimization, and symmetry-aware modeling, this framework enables a direct study of NH many-body systems without Hermitian embeddings or adiabatic continuations.