A not entirely formal discussion of some aspects of the so called PT-symmetric quantum mechanics will involve the following four more or less open problems: 1. What are chances of a closed-form description of boundaries of stability (i.e., of the allowed domain D of physical parameters) in a generic PT-symmetric model? (hint: we shall demonstrate that they are better than expected) 2. What happens if the coordinates are allowed to be topologically nontrivial? (hint: one gets the so called quantum toboggans) 3. Should and could a realistic PT-symmetric model be also based on a non-Hermitian form of the parity? (we intend to recommend it) 4. Could the use of a realistic PT-symmetric model be made compatible with a time-dependent metric? (we shall show how).