I have a GTX1080, but I'd be getting an RX-480 8GB if that was my budget.
Worth watching:
https://www.youtube.com/watch?v=gEw3CaNSbUo
The game consoles have AMD GPU's that are nearly identical so the code is probably going to be more optimized towards these GPU's as time goes on (if intrinsic shaders for DX12/Vulkan are similar low-level code get implemented for console they probably will work fine with the PC version). There is some CONFUSION about NVidia DX12 performance. It's not that they "don't support DX12" but more like their DX11 drivers are using most of their GPU more efficiently.
There are other reasons, but AMD with DX12/Vulkan is partly pulling ahead because they weren't fully utilizing the GPU. Again, you can't increase the frame rate if the GPU is already fully used. However, that's actually GOOD NEWS since the RX-480 is slightly cheaper so we could see 10% to 20% improvements in some titles in the near future vs a similar GTX1060 but that's not certain.
DOOM is a good indicator though and AMD is kicking ass there. Can NVidia do some tweaks to improve as well? Yes, but the advantage still swings to AMD if only due to some untapped potential.
Basically the RX-480 is slower in most current and past games, but is pulling ahead in DX12/Vulkan. Since FUTURE games are more demanding than PAST games I'd go with the RX-480 for this reason.
The Asus Strix is my top choice.
You also get 8GB of VRAM which the GTX1060 6GB and 3GB models obviously don't have. Some games are using a LOT of video memory already, and it will only increase.
My top choice (about $250 to $270USD)
http://pcpartpicker.com/product/hkJkcf/asus-radeon-rx-480-8gb-rog-strix-video-card-rog-strix-rx480-o8g-gaming