Just like the movie, a being starts following you at a certain point, but it can't be killed, walks really slowly and constantly tracks your location. And it will take on the appearance of a character in the vicinity that you trust, or even a common bad guy. You basically just have to avoid it but if you go into an enclosed location it might make for a fun game experience when it catches up to you and you are in the middle of a firefight. Oh and it kills you after 2 or 3 hits so you have to get away.

Might sound stupid but I think it would be pretty cool to see that thing pop up every now and then when you're playing the game and it would add a new dimension of horror as well.

It might even take the form of really weird looking monsters if you're down in a dungeon at some point.

And if you fast travel it stalks you but might only travel at 1/4 the speed so it won't just be right behind you when you fast travel somewhere.

Oddly enough.... there was a thread a few weeks ago that described something of that nature. It wouldn't attack or interact, it was invisible and was hard to detect. Most people began to notice " it " when swimming as it made a wake in the water. I cant link because im on mobile but I believe it was posted in the general discussion forum for FO 4.