  We study the families of nonlinear modes described by the nonlinear
Schr\"odinger equation with the PT-symmetric harmonic potential $x^2-2i\alpha
x$. The found nonlinear modes display a number of interesting features. In
particular, we have observed that the modes, bifurcating from the different
eigenstates of the underlying linear problem, can actually belong to the same
family of nonlinear modes. We also show that by proper adjustment of the
coefficient $\alpha$ it is possible to enhance stability of small-amplitude and
strongly nonlinear modes comparing to the well-studied case of the real
harmonic potential.
