Stochastic jump phenomena in the random responses of a Duffing oscillator subjected to narrow band excitation are investigated. The stochastic jump phenomena correspond to the existence of multiple stationary responses, which differ in phase to the excitation. In our previous paper, the product of each real wavelet of response and excitation is proposed as the phase of each frequency component. But taking the moving average of the product over one period is necessary because of pulsation. In this paper, the product of the complex wavelet transform of the response and the complex conjugate of that of the excitation is used to evaluate phase of each frequency component. The central value of unstable region of the phase in the frequency response function is used as the center of the range of the principal value of the complex argument, and the value and bounds of unstable region are taken as the threshold for phase obtained from the wavelet transforms. Using this range of principal value of complex argument and thresholds, two states of the response are successfully identified, as well as the previously proposed criterion using real wavelet transform.