  We propose and investigate several complex versions of extensions and
restrictions of the Skyrme model with a well-defined
Bogomolny-Prasad-Sommerfield (BPS) limit. The models studied possess complex
kink, anti-kink, semi-kink, massless and purely imaginary compacton BPS
solutions that all have real energies. The reality of the energies for a
particular solution is guaranteed when a modified antilinear CPT-symmetry maps
the Hamiltonian functional to its parity time-reversed complex conjugate and
the solution field to itself or a new field with degenerate energy. In addition
to the known BPS Skyrmion configurations we find new types that we refer to as
step, cusp, shell, and purely imaginary compacton solutions.
