Been thinking a lot about my pitch to become the next Dr Who show-runner. I would like a historical Quaker companion because of who I am as a person and also because I think it would be good for it.
I think one of the main reasons Jamie is so compelling as a companion is because he has a distinct ideology that's fundamentally shaped his life so far. Not only this, but it's one that's mostly unfamiliar to modern audiences so it's not going to fall into trite expectations, and it can interact with the story in interesting ways, particularly in the different choices that he and the Doctor make.
Even though they definitely didn't make much of it in the show itself, I think it's been fertile ground for transformative works. This shows there are still good stories out there to be told that haven't been told by Dr Who as a show yet! Even better, they're stories told through a companion's perspective, strengthening an overall arc.
A historical Quaker companion brings in fun quirks about how you adapt a belief system that is distinctly of a certain time and place to other times and places. What do they wear? How do they worship? Will they celebrate in the obligatory Christmas special? This can be fun, particularly if you have a modern(ish) companion to play against; they can even act a foils so we can see that the modern companion's choices are just as guided by culture and context. (Yes, my favourite Team TARDIS is Polly, Ben and Jamie.)
The best thing about a Quaker companion is that they would be a real stick in the mud. The Doctor needs them to show deference to a Lord? It's not going to happen! Most notably, Dr Who has always flirted with questions of pacifism and I think having a religiously motivated pacifist character would force it to actually grapple with them.
(I concede that they would be a real stick in the mud for the writing team too, but I think that Dr Who badly needs the challenge of constraints, particularly regarding consistency.)
Ultimately, my favourite series is still New Who S1 with Rose because it's the one that most completely asks and answers the question, what would it be like to leave your life to travel through space and time and also have to deal with Situations.
I want to understand the concessions you make, and which you don't. How are your beliefs challenged and how are they confirmed? Do you act how you thought you would when you imagined a hypothetical? How do you act in a situation you never hypothesised? Who do you make friends with? How are you changed - is it beyond recognition? Can you go home again?