[math-fun] The speed of wave motion

For wave = sin(A*x - B*t) with A,B real constants, x=space, t=time, we all agree the speed of wave motion is |B/A|. But now consider exp(i*A*x - i*B*t) where A,B could be permitted to be complex. What would you claim the "speed of wave motion" is now? "Speed of 'information transmission'?" I'm not sure there is a clear and right answer... but send yours...