SNR for the magnitude of a complex signal
The magnitude $|y[n]|$ follows a Rice (aka Rician) PDF. You can find analytical expressions for the average, variance and SNR in Voltage Signal-to-Noise Ratio (SNR) Nonlinearity Resulting From Incoherent Summations and in The Squaring-Loss Paradox. These two references further investigate the SNR of the addition of independent magnitudes.