  We show that complex Lie algebras (in particular sl(2,C)) provide us with an
elegant method for studying the transition from real to complex eigenvalues of
a class of non-Hermitian Hamiltonians: complexified Scarf II, generalized
P\"oschl-Teller, and Morse. The characterizations of these Hamiltonians under
the so-called pseudo-Hermiticity are also discussed.
