Can I display a single PS4 console on 2 different game monitors for the purpose of multi-player gaming (split-screen over 2 monitors)?

What you really want is for the game to not be split screen anymore - you want one player on screen 1, full screen, and one player on screen 2, also full screen.

Unfortunately, this isn't going to be possible. There's no support in the PS4 for this, and there's no support in the game for this either. The PS3 had something called "SimulView" but this isn't supported in PS4 games to date.

There are various solutions external to the PS4, but the problem is that the aspect ratio of each player's portion of the screen won't match up with the full size of each display. For instance, a link in the comments talks about using a passive 3D display (assuming your TV or TVs support passive 3D) to get a full screen image out of a split screen game.

The problem here is that the game is still only sending half a screen's worth of pixels to each player, and then that image is stretched to fill the entire screen. It's going to look very out of proportion - HUD elements will look double height at the same width, etc.

Thus, there aren't any better options than just running the console to a HDMI splitter and displaying the same split-screen image on both TVs.

The only way to get exactly what you want is to pick up a second PS4 for your second TV.