I genuinely think a Kilgrave/Doctor ship would work. The Doctor constantly ends up in love with Insane People That Will Kill™ (the Master, canonically considering twelve's era), while Kilgrave has mainly been obsessed with a superhuman hero (Jessica, obviously).
Fun thing is that I think they could possibly end up going along together, if the Doctor was immune to mind control, of course (something that might be likely, he's a Time Lord, with a different more advanced biology, seeing as the mind control is actually biological, it might have a completely different or no effect). It would just be a nice circle of the Doctor going 'I can fix him' and Kilgrave tagging along if he actually gets obsessed, while absolutely hating the adventures (he's like a psychopath city Karen with mind control, you can't convince me he'd like particularly big trips). They are perfect for each other.
