We investigate both the quantum and classical dynamics of a non-Hermitian system via a kicked rotor model with PT symmetry.For the quantum dynamics,both the mean momentum and mean square of momentum exhibit the staircase growth with time when the system parameter is in the neighborhood of the PT symmetry breaking point.If the system parameter is much larger than the PT symmetry breaking point,the accelerator mode results in the directed spreading of the wavepackets as well as the ballistic diffusion in momentum space.For the classical dynamics,the non-Hermitian kicking potential leads to the exponentially-fast increase of classical complex trajectories.As a consequence,the imaginary part of the trajectories exponentially diffuses with time,while the real part exhibits the normal diffusion.Our analytical prediction of the exponential diffusion of imaginary momentum and its breakdown time is in good agreement with numerical results.The quantum signature of the chaotic diffusion of the complex trajectories is reflected by the dynamics of the out-of-time-order correlators(OTOC).In the semiclassical regime,the rate of the exponential increase of the OTOC is equal to that of the exponential diffusion of the complex trajectories.