  The theory of non-Hermitian systems and the theory of quantum deformations
have attracted a great deal of attention in the past decades. In general,
non-Hermitian Hamiltonians are constructed by an ad hoc manner. Here, we study
the (2+1) Dirac oscillator and show that in the context of the
$\kappa$-deformed Poincar\'e-Hopf algebra its Hamiltonian is non-Hermitian but
has real eigenvalues. The non-Hermiticity stems from the $\kappa$-deformed
algebra. From the mapping in Bermudez et al., Phys. Rev. A 76, 041801(R)
(2007), we propose the $\kappa$-Jaynes-Cummings and
$\kappa$-anti-Jaynes-Cummings models, which describe an interaction between a
two-level system with a quantized mode of an optical cavity in the
$\kappa$-deformed context. We find that the $\kappa$-deformation modifies the
Zitterbewegung frequencies and the collapses and revivals of quantum
oscillations. In particular, the total angular momentum in the $z$ direction is
not conserved anymore, as a direct consequence of the deformation.
