Thing I read way too much into re: the Doctor and Clara’s relationship in Deep Breath:
During the 11th Doctor′s regeneration, they very pointedly remind the audience that the Doctor imprints on people after regenerating (having him remember Amy and explain aloud the significance of her being the first face he saw) and then the very first thing the 12th Doctor does is peer into Clara’s face with deep deliberation.
And, on Clara’s side of things, it understandably freaks her out a bit, right? A minute ago he was this guy who, sure, he’s an alien, but the exact brand of “not human” 11 would display to people was usually endearing and quirky and palpable… he buried the stuff that wasn’t as best he could. But suddenly there’s 12, and he’s this unabashedly alien presence, not trying to be young or human anymore… his gaze here is very unreadable, animal-ish. He’s different now in ways that are very hard for Clara to parse, and it’s evident before he’s even said a word. Her part of Deep Breath is about learning to understand him as the same person despite his sudden shift in personality and behavior.
However, from the Doctor’s perspective:
We find out in Deep Breath that 12 is the way he is because, subconsciously or no, he trusted Clara to see him as the blunt, inhuman, unfathomably old mess the Doctor really is and still accept him for it:
VASTRA: He looked young. Who do you think that was for?
CLARA: Me?
VASTRA: Everyone. I wear a veil as he wore a face; for the same reason.
CLARA: What reason?
VASTRA: The oldest reason there is for anything. To be accepted.
(…)
VASTRA: The Doctor regenerated in your presence. The young man disappeared, the veil lifted. He trusted you. Are you judging him?
So I think that it’s very significant to him that she’s the first person he sees here. He’s making sure he looks at her properly and really sees her. He’s now this person who might not be acceptably human anymore, and he needs the face he imprints on to be someone he trusts to know him and see him despite that. It’s going to define him.
But, again, his whole persona and appearance pretty visibly freaks her out. All he sees throughout Deep Breath is her looking scared and upset, all he hears is her talking to Jenny about how he’s not the person she cares for and how she wants him to change back (to which he mumbles in his sleep about not being seen). It’s a scary moment of truth for him, where his friend has told him that, for all his trust, he really isn’t acceptable without that veil. While she’s struggling to see him as the Doctor, he’s struggling with this sense of abandonment, and at the end he breaks down as much as he’s liable to and asks that she not leave him like that:
DOCTOR: You can’t see me, can you? You look at me, and you can’t see me. Have you any idea what that’s like? I’m not on the phone, I’m right here, standing in front of you. Please, just, just see me.
And after she’s had time to think about what that first scary glare towards her actually meant, what does she do to show him that she’s going to make that effort for him? She mimics his imprinting, and looks at him with the same careful consideration as he gave her.
Shows him that, even though this gesture is entirely non human, she gets it and understands the trust behind it and actually does see him.
