  A few recent innovations of applicability of standard textbook Quantum Theory
are reviewed. The three-Hilbert-space formulation of the theory (known from the
interacting boson models in nuclear physics) is discussed in its slightly
broadened four-Hilbert-space update. Among applications involving several new
scattering and bound-state problems the central role is played by the models
using apparently non-Hermitian ("crypto-Hermitian") Hamiltonians with real
spectra. The formalism (originally inspired by the topical need of
mathematically consistent description of tobogganic quantum models) is shown to
admit certain unusual nonlocal and/or "moving-frame" representations of the
standard physical Hilbert space of wave functions.
