The Last of Us was fantastic, mechanically it made me pretty uncomfortable in terms of gameplay with the aggressive violence. The story is also top notch.
Bioshock Infinite had unsettling moments due to the subtle racial undertones and crazy religious aspects of its story.
The Wolf Among Us- because it's clearly the best thing TellTale Games has ever made.
Spec Ops: The Line- I can't even really talk it about without spoilers. Except, emotional trainwreck.
If you enjoyed Max Payne 1 & 2 then I suggest 3 just to see what sort of state Max is in now.
Inversion- It had interesting cover mechanics due to situations with zero gravity(dead space) you also play as a cop in the middle of an invasion looking for his wife and daughter.
@bluefish recommended Devil May Cry(2013) and The Darkness. DmC had some brilliant, subtle, complex acting that reminded me of my old theater days. The story was also an interesting critique on society today. I much preferred the Darkness 2 over the first one, it had strong themes of loss, regret, and strength that really stuck with me. Though the last hour of the first game sums itself up perfectly. So both are good.
Remember Me- Because that was a surprise left field from Capcom, and it was genuinely honest and kind of sad in a weird way. Poor Nillian.
I found Infamous 2 emotional, at least in the first half of the game because your the ONLY superhero and the world is basically fucked.
Lost Planet 3- another Capcom surprise. If you liked Dead Space, then it's slightly similar but the main character is way more relate-able. The writing feels really natural and it also gets suuuper creepy, in a really dark way. And lastly...
Resistance 3. Really, it's fantastic.
^ Resistance 3, it really doesn't get enough love.
Hope it helps! Happy gaming.