I do like her and Six though, she gives about as good as she gets and I a bit ship it, it's like S8 of Nu Who but more rainbow.
This! I sort of ship it too (which took me VERY by surprise, I don't ship Dr/Companion), but like you say there is very much a Twelve/Clara vibe - they're not really good for each other, and they are not really able to show how they feel and they argue a lot, but they DO care very very deeply about each other. I think it's the fact that Six freaks out when Peri is threatened the same way Twelve freaks out over Clara. 'If she dies I WILL END YOU!'