Anomalous mechanisms of the loss of observability in non-Hermitian quantum models

•A new mechanism of quantum phase transitions proposed.•The conventional one, via an isolated exceptional point (EP), generalized.•The general horizon of a loss of observability interpreted as a confluence of EPs.•Innovation illustrated via several non-Hermitian toy-model Hamiltonians.•Constructions based on linear algebra kept non-numerical. Via several toy-model quantum Hamiltonians H(λ) of a non-tridiagonal low-dimensional matrix form the existence of unusual observability horizons is revealed. At the corresponding limiting values of parameter λ=λ(critical) these new types of quantum phase transitions are interpreted as the points of confluence of several decoupled Kato's exceptional points of equal or different orders. Such a phenomenon of degeneracy of non-Hermitian degeneracies seems to ask for a reclassification of the possible topologies of the complex energy Riemann surfaces in the vicinity of branch points.