So I just spent the past several hours trying to get my PS4 to cooperate with my HDMI-DVI adapter on my PC monitor--and won! I hear it's a common problem for monitor users. Ran back and forth to my living room HDTV and monitor so many times, thinking I'd have to use the TV to set lower and lower resolutions to get the adapter to recognize, but the solution's actually pretty simple.
Basically what he's saying: there's a bug in the PS4. To get past it, have the PS4 initially off, then hold the power button down until the system beeps twice. This puts it into Safe Mode at 480p. Somehow this makes the screen appear, when it very much would not with the HDMI cable alone.
It will give you a list of options, one in which you can tell it to restart and readjust the PS4's resolution again. The system restarted, and then I just went to the system display as I normally would on a TV to change it to 720p (my DVI adapter's threshold.)
With fingers crossed, it worked. It even worked after shutting the system down and booting it up again.
So if anybody wants to connect to their monitor in the future, that's how you do it. During all of this, I tried turning off the HDCP to see if that would help--kind of scared to turn it back on.