Classical oscillator with position-dependent mass in a complex domain

We study complexified Harmonic Oscillator with a position-dependent mass, termed as Complex Exotic Oscillator (CEO). The complexification induces a gauge invariance [A.V. Smilga, J. Phys. A 41 (2008) 244026, arXiv:0706.4064; A. Mostafazadeh, J. Math. Phys. 43 (2002) 205; A. Mostafazadeh, J. Math. Phys. 43 (2002) 2814; A. Mostafazadeh, J. Math. Phys. 43 (2002) 3944]. The role of PT-symmetry is discussed from the perspective of classical trajectories of CEO for real energy. Some trajectories of CEO are similar to those for the particle in a quartic potential in the complex domain [C.M. Bender, S. Boettcher, P.N. Meisinger, J. Math. Phys. 40 (1999) 2201; C.M. Bender, D.D. Holm, D. Hook, J. Phys. A 40 (2007) F793, arXiv:0705.3893].