Does it have to be on the PS4?

What about the F2P Team Fortress 2?

DOOM series, Wolfenstein series, Battlefront series, Dying Light, Left 4 Dead, Team Fortress 2, Killing Floor 1-2, Planetside 2, RAGE, Contagion, Borderlands series.

Call of doody

Wolfenstien the new order i suppose. But if you have a PC or PS3, FEAR is one of my favourites and FPS is the worst genre in my opinion, it does something right.

Perfect Dark (N64). This game is still my all-time favorite FPS. If you don't have an N64, but have an X360, the XBLA port is pretty good too, although the controls bugged me a bit.

If you're more interested in campaigns, then I'd recommend Wolfenstein: the New Order, Metro Redux or Shadow Warrior. If you're more interested in multiplayer, then you probably want to go for something like Battlefield 4, Call of Duty: Black Ops III or Destiny.

Not a fan of FPS so i can't say, but if you considered Fallout 4 an FPS as well then yes. Def get Fallout 4 :) !!

Bioshock Infinite

Battlefield 4 is really cheap right now. It's like $18 for the premium version on the PS4 right now.

Out of all those games, I think in terms of competitive multiplayer black ops is your best bet. I think battle front will get boring real quick imo.