Not sure what you are asking here. If you are asking about eigenvalue solvers, a non-symmetric solver is generally needed when losses are present. Eigenvalues will appear in conjugate pairs. Acoustic radiation in all fluids is dominated by pressure waves. Viscous shear forces and thermal diffusion contribute to loss (but may not be significant in your problem). Correct handling the boundary condition between fluid and cantilever is critical to a good solution. Hope this helps.
I believe that the real part of the complex frequency is the resonant frequency, and the imaginary part is the damping.
If you plot the imaginary part against time, it will form a spiral on the complex plane. The distance between two maxima of the real part will be the period, which for small damping will be 1/f. I am not sure if your frequency is the radian frequency (omega=2*pi*f) or the cycle frequency (f). The complex amplitude is exp(j*2*pi*f(complex)*t)=exp(j*2*pi*f(real)*t)*exp(-2*pi*f(imaginary)*t).