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Extended quantum mechanics using non-Hermitian (pseudo-Hermitian) Hamiltonians H = H‡ is briefly reviewed. A few related mathematical experiments concerning supersymmetric regularizations, solvable simulations and large-N expansion techniques are summarized. We suggest that they could initiate a deeper study of nonlocalized structures (quasi-particles) and/or of their unstable and many-particle generalizations. Using the Klein-Gordon example for illustration, we show how the 𝒫𝒯 symmetry of its Feshbach-Villars Hamiltonian HFV might clarify experimental aspects of relativistic quantum mechanics.