From the times of Herman Feshbach (b. 1917) quantum theory of open systems and of the non-unitary evolution of resonant states works with complex energies and non-Hermitian effective Hamiltonians. Thanks to the six years younger Freeman Dyson (b. 1923) it is known, nevertheless, that the implication only goes in one way. Non-Hermitian Hamiltonians can have real spectra and generate unitary evolution of isolated quantum systems. In the talk, the essence of the apparent paradox will be explained. A compact review will be given of the recent growth of interest in the latter class of non-Hermitian Hamiltonians. Two (viz, stationary and non-stationary) versions of the resulting innovative 3HS formulation of quantum theory will be reviewed, and its innovative appeal will be illustrated by a few examples involving the analytically continued Bose Hubbard multibosonic model and the stationary and non-stationary versions of the first-quantized Klein-Gordon field.