  Simple examples of non-Hermitian Hamiltonians with purely real spectra
defined in $L^2(R^+)$ having spectral singularities inside the continuous
spectrum are given. It is shown that such Hamiltonians may appear by shifting
the ndependent variable of a real potential into the complex plane. Also they
may be created as SUSY partners of Hermitian Hamiltonians. In the latter case
spectral singularities of a non-Hermitian Hamiltonian are ordinary points of
the continuous spectrum for its Hermitian SUSY partner. Conditions for
transformation functions are formulated when a complex potential with complex
eigenenergies and spectral singularities has a SUSY partner with a real
spectrum without spectral singularities. Finally we shortly discuss why
Hamiltonians with spectral singularities are `bad'.
