GTX960 4GB.
http://pcpartpicker.com/parts/video-card/#c=208&r=4096&sort=a8&page=1
Recommended:
http://pcpartpicker.com/part/evga-video-card-04gp43967kr
I have a hard time recommending AMD in general now:
a) poor DX11 drivers (the worse the CPU the more likely the bottleneck will be)
b) missing features like PhysX (may not care, but in a few games I find it worthwhile. In others it's off or low)
c) Power consumption (may not care but it's a big deal in my room to avoid added heat)
Performance comparison:
SIMILAR depending on the game: http://www.kitguru.net/components/graphic-cards/luke-hill/sapphire-r9-380-nitro-4gb-review/7/
The GTX960 also supports MFAA (supported games) for higher frame rate. Pretty sure it overclocks better than R9-380.
DX12 gaming performance?
I can't comment until a few games are out. Yes, AMD has asynch compute, but how much the average game will benefit is an unknown.
Finally, AMD's support in the future is uncertain. They are suffering financially, and it's unknown how this will affect their ability to produce TIMELY DRIVER UPDATES for new games or add/support other features.
Summary:
I'm having a hard time recommend anything other than a GTX960 4GB card for this budget.