No, I said if you want to do 1080p at 144hz, you can use Dual Link DVI, HDMI 1.3 or Displayport. All three connection types can do 1080p at 144hz, but beyond that resolution and frequency you'd want go with Displayport.
I never said anything about all 144hz monitors taking all three types of connectors. I said the actual output of the video card and the inputs the monitor had would determine what kind of cable would need to be used.
And it's just fantastic that your 1080P 144hz monitor only takes Displayport. But it's not the only one on the market and other monitors may take other inputs. But really that's that's neither here nor there.
The OP asked about if DVI or HDMI is the best connection to use, and frankly saying use Displayport doesn't answer that question. Is Displayport the better cable to use? Yeah probably, but only if all the equipment in question can use it, and we have no monitor to use as reference to determine if thats an option. Hence the whole "inputs the monitor takes that'll determine what kind of cable you need" part of my statement.
The OP also asked for monitor recommendations, which would have helped narrow the scope of the whole discussion, but you and I were too busy bickering over what amounts to the having the same answer to actually do anything with that.
So, to sum all this nonsense up; if the OP only wants to do 1080p at 144hz, HDMI 1.3 or Dual Link DVI will give you the same performance. Displayport will also work just fine in this case as a 3rd option. But keep in mind, your monitor will determine the exact cable(s) you are able to use.
Beyond the scope of the OP's question. If you want to do higher than 1080p then you'll want to go with Displayport. It's more common on monitors 1440p and higher. HDMI 2.0 will also work for 1440p at 144hz but it's a less common option than Displayport. And once again, your monitor will determine the exact cable(s) you are able to use.
If the OP wants to have a better video playback experience, and the monitor supports it (like xSimply1337x mentioned), then the OP will want to stick to HDMI as it has RGB as well as YCbCr support. Video (DVD, BD, HD DVD, cable, blah blah blah) is YCbCr, so for video playback HDMI will produce better black levels and accurate color. However if all the OP cares about is gaming, then it's a non issue, as PCs are RGB and so HDMI and DVI will effectively reproduce color the same with that content.