@Kodachrome2000, I would wait until you have your HD-600 first before thinking about an amp or DAC change. If you like to let out your "bass-head" every once in a while I think the HD-650 would be a better fit but since you want more neutral/natural then the HD-600 is a better choice.
I had the HD-598 for a while before giving them away and like
@Midgetguy I was using a Topping VX1 DAC/amp and it was great for the price but the HD-600 needed a slightly better amp to make it sound like an "upgrade" from the HD-598, and as another poster suggested tubes is where it shines.
If wanting to keep the natural/neutral sound some say the Valhalla2 is good, for me its a BH Crack or Ember and depending on the tubes used it will "color" the sound somewhat, not something I would consider drastic but it is noticeable. For flexibility the Ember is hard to beat while the BH Crack is limited to higher impedance headphones like the HD-600/650/800 but it really shines with them.
For playing games I really like the HD-700, much better sound stage, positional accuracy, and dynamics, its also much easier to drive properly than the HD-600.