  It is by now well established that the momentum space dual to the
non-commutative $\kappa$-Minkowski space is a submanifold of de Sitter space.
It has been noticed recently that field theories built on such momentum space
suffer from a subtle form of Lorentz symmetry breaking. Namely, for any
negative energy mode the allowed range of rapidities is bounded above. In this
paper we construct a complex scalar field theory with a modified action of
Lorentz generators which avoids this problem. For such theory we derive
conserved charges corresponding to translational and U(1) symmetries. We also
discuss in some details the inner product and Hilbert space structure of the
$\kappa$-deformed complex quantum field.
