For a first crack at doing a psychological exploration of the movie, consider its very title:
Prisoners. In my interview with the movie's screenwriter Aaron Guzikowski, I asked him about the importance of the title:
Just the idea it obviously has the double meaning of it’s about this guy who takes this captive and holds him prisoner, and obviously people are kidnapped and they’re kept prisoner, but it’s also that these characters are prisoners of their own psyche, their own mental decision making.
So not only the kidnapping of the girls, but also the turnaround of Keller taking Alex hostage, that's pretty obvious. But what about "characters are prisoners of their own psyche"?
We already looked at Keller in this light the other day which you can read about
here. Basically Keller is held prisoner by his survivalist mentality. Remember this quote: “The most important thing your grandfather taught me is to be ready.” Yet he was NOT ready and 'allowed' his daughter to be kidnapped. So at one level, he is imprisoned by his high expectations to be a protective father and husband.
That opens up another layer of his psychological imprisonment: His rage. Keller is at his core a violent man. We see this emerge as his carefully constructed life and persona crumbles over the course of the story. It makes me wonder if one reason he quit drinking in his life leading up to FADE IN had something to do with anger control issues. Just pure speculation. In any event, that rage is there. It fuels his obsession with Alex - along with his sense of shame at not being ready enough to protect his daughter - and leads him into his downward psychological spiral in which he is willing to torture and nearly kill another human being.
Those two emotions - shame and rage - have him shackled and overtake his logic. So it is ironic on multiple levels that he himself becomes a LITERAL prisoner at the end of the story.
There's another way in which this theme of being a prisoner works and that's with Loki. Anyone care to take a crack at that?