Jungian interpretation - SPOILERS
For me the whole "unique" element, so clearly stated at the film's end and referenced earlier, is at the heart of what's happening in "The Double". Of course this movie is kafkaesque, and existential (and many other things), complete with an existentialist leap off the ledge, but I also see a Jungian process going on. Last week Jesse Eisenberg was on David Letterman and he talked about the doppelganger as kind of being the Jungian idea of "the shadow"; this I feel really does apply overall to the film. The whole idea of "the process of individuation", which is the central concept of Jung's psychology, is the expression of this unique individuality; it's making what's unconscious, conscious - all the repressed, darker, unacceptable parts of the individual's hidden side. This describes the idea of "the shadow", which is a good description of the doppelganger, James. This shadow has to be confronted and integrated, rather than just projected onto the outside world, in order for the person to give birth to his unique identity. It's a form of self-healing. I see Hannah as being an expression of the "anima", which is the other great Jungian archetype that is confronted when struggling toward individuation. In the case of a male, he has to confront his unconscious feminine aspect (women have the "animus") to achieve his integration. The anima can have both a positive, creative aspect or a negative, destructive aspect - she can be a creative muse and inspire, or a siren, a kind of nymph leading men to their death. These are the projections a male places on a particular woman in order to bring out hidden aspects of his own personality, his repressed feminine side. Hannah, from the perspective of Simon, has both a kind nurturing side that inspires him, and a side that's judgemental and guiding him to destruction, to suicide. This is how Simon sees her, and Hannah is extremely mysterious and ambiguous in the film, perhaps because she's reflecting these opposite aspects of the anima. Simon swings between love (tenderness and empathy) and obsession (the telescope, the garbage bin), which are reflections of the positive and negative pull of the anima. An interesting note is that I liked how they showed Hannah creating her own artwork out of her blood (drawing her own double reflections) to me this indicated that this character had her own process of individuation, her own reality, and was making her own projections onto others (something we all do) - she, too, was like Pinocchio and was longing to feel "real" - to feel unique.
To me the film is very dreamlike and, as in a dream (and in waking life, too) our experience of the inner-outer world depends on fantasy-images of the psyche that we project. Our imagination creates these images and shapes the material realities that we experience. It's kind of like an artist creating his work out of his imagination. For me "The Double" expressed this beautifully, as we were seeing a reality created in the mind of Simon, a non-person, giving birth to himself.