  In a special representation of complex action theory that we call
``future-included'', we study a harmonic oscillator model defined with a
non-normal Hamiltonian $\hat{H}$, in which a mass $m$ and an angular frequency
$\omega$ are taken to be complex numbers. In order for the model to be sensible
some restrictions on $m$ and $\omega$ are required. We draw a phase diagram in
the plane of the arguments of $m$ and $\omega$, according to which the model is
classified into several types. In addition, we formulate two pairs of
annihilation and creation operators, two series of eigenstates of the
Hamiltonians $\hat{H}$ and $\hat{H}^\dag$, and coherent states. They are
normalized in a modified inner product $I_Q$, with respect to which the
Hamiltonian $\hat{H}$ becomes normal. Furthermore, applying to the model the
maximization principle that we previously proposed, we obtain an effective
theory described by a Hamiltonian that is $Q$-Hermitian, i.e. Hermitian with
respect to the modified inner product $I_Q$. The generic solution to the model
is found to be the ``ground'' state. Finally we discuss what the solution
implies.
