Does the ammonium ion hydrogen bond with water?
There is a detailed consideration of this issue in  Hydrogen Bonding of the Ammonium Ion Journal of Molecular Structure 345 (1995) 77-81.

First, the try to be clear about what they mean by "hydrogen bond":


  For the purposes of the discussion, a hydrogen
  bond is taken to be a specific short-range interaction
  in which the N-H bonds within the
  ammonium ion are significantly affected by the
  surrounding ions, and not simply electrostatic
  attraction.


They conclude that ammonium ions rotate too freely in water for significant hydrogen bonding to exist.  

In gas phase calculations, they find that one water molecule hydrogen bonds to one ammonium ion (in other words, the dimer is hydrogen bonded), based upon the lengthening of the N-H covalent bond, but as you add 2, 3, and 4 water molecules, the N-H bond length returns to the free ammonium N-H bond length and the O --- H length increases.