  Following a generic approach that leads to Bogomolny-Prasad-Sommerfield (BPS)
soliton solutions by imposing self-duality, we investigate three different
types of non-Hermitian field theories. We consider a complex version of a
logarithmic potential that possess BPS super-exponential kink and antikink
solutions and two different types of complex generalisations of systems of
coupled sine-Gordon models with kink and antikink solution of complex versions
of arctan type. Despite the fact that all soliton solutions obtainedin this
manner are complex in the non-Hermitian theories we show that they possess real
energies. For the complex extended sine-Gordon model we establish explicitly
that the energies are the same as those in an equivalent pair of a
non-Hermitian and Hermitian theory obtained from a pseudo-Hermitian approach by
means of a Dyson map. We argue that the reality of the energy is due to the
topological properties of the complex BPS solutions. These properties result in
general from modified versions of antilinear CPT symmetries that relate
self-dual and an anti-self-dual theories.
