[Henry:] When I am out there, in time, I am inverted, changed into a desperate version of myself. I become a thief, a vagrant, an animal who runs and hides. (Prologue.11)
Cast into a different time, Henry is stripped of his present identity and becomes a wild animal, struggling for survival.
[Clare to Henry:] "Yes… you're more…" selfish, I think, but I say, "…younger." (1.1.137)
Clare somewhat admits to Henry that he's not the man she knows from her memories. His younger self seems more flawed and not quite as emotionally mature.
[Henry to Henry:] "Can't we just look at the art?" pleads Henry. He's nervous. He's never done this before. (1.4.2)
Henry's younger self is still very much a nine-year-old good boy, raised to think of stealing as a crime. He hasn't yet experienced a situation that would force him to value his survival over his moral standards.
[Henry:] A translucent moment. I didn't understand and then I did, just like that. I watch it happen. I want to be both of us at once, feeling again the feeling of losing the edges of myself, of seeing the admixture of future and present for the first time. (1.4.42)
Henry witnesses his younger self realizing that they are actually the same person. He wishes he could go back to that moment, longing to unite his two selves into one person.
[Henry:] She is completely impassive, cool as a Raymond Chandler murderess. […] Her laugh sounds spooky, echoing through the woods […] and it seems to me that this moment is the demarcation, a sort of no-man's-land between Clare's childhood and her life as a woman. (1.5.214-225)
Henry is taken aback by how much Clare enjoys humiliating Jason for hurting her. He discovers a new violent side in Clare, but attributes it to her fall from the innocence of her childhood to her new strong, independent-thinking identity as a young woman.
[Henry:] What would I be without [Clare]? Not the man who breathes, slowly, deeply across the bed from me. (1.8.232-233)
Henry realizes that his sense of self is closely tied to Clare. Being with Clare grounds him in his present life and allows him to find temporary peace from his disorder.
[Gomez to Henry:] "Are you the same person?" (1.14.91)
Gomez thinks that Henry's condition makes him a flawed being with no sense of identity that can stick around through all the traveling.
[Henry:] Whatever I am. What am I? What am I? (2.2.231)
Henry questions his identity. Is he an advanced human being, or is he a faulty version of a human being? What do you think?
[Henry:] There is a noise in the shrubbery. I open the window farther and look down. There I am, sitting in the dirt, under the window, soaking wet, panting. He grins at me and gives me the thumbs up. (2.14.45)
Henry deals fairly easy with meeting other versions of himself. But in which version does his sense of identity and consciousness reside at any given time? Or does he split his identity between various versions and how might that feel?
[Clare:] I regard my likeness and she returns my gaze. I place my finger on her forehead and say, "Vanish," but it is she who will stay; I am the one who is vanishing. (3.4.6)
After Henry's death, Clare draws an image of herself to assert her existence in this life, to create an identity that lasts. She knows that it's her drawing – this physical evidence – that will live on when she passes away.