Unless the alternator bearing being dead and off center has somehow damaged your power steering bearing, the two are completely seperate systems... (I highly doubt this happened as well)
I would think you just have another problem now... The fluid being there has nothing to do with the bearing in the pump dying... It has just passed its useful life.... That being said, my power steering pump is dying too, and makes the same noise - only on cold starts though... I have fluid at the right level as well.... Its just time for a replacement!!!