Each new companion in Doctor Who is a reaction to the previous inhabitant of the TARDIS guest bedroom. Amy and Rory Pond/Williams offered the Doctor a chance to experience the growth and support of family life after the Tenth Doctor's traumatic spell of going it alone, and when they were taken from him by the Weeping Angels, it took an impossibly odd situation with Clara Oswald — an enigma wrapped in a puzzle wrapped in a Dalek shell — to encourage him to take a chance on friendship again.

Bill Potts (played by Pearl Mackie) arrived in the Twelfth Doctor's life at a time when he was attempting to stop being the intergalactic traveler, when he was a college professor with a dark secret locked away in a vault. He'd forgotten Clara, but still felt the void of her presence, and with Nardole being relatively unflappable as an assistant, he needed someone to show off to. But not just anyone, the Doctor needs approval from people who only give it grudgingly, which is presumably why so many of his companions are relatively young and stroppy by human standards.

Bill may be young, but she's not inexperienced. She's a natural underdog. She serves chips to privileged students in Bristol University's canteen, while secretly taking advantage of the educational opportunities they take for granted. She has had to plow her own furrow, partly from being underestimated thanks to her race, her class and her sexuality, and partly the loss of her mother. But, and this is key to why the Doctor likes her so much, she's done it without losing the thrill of acquiring new information. She reflects the joy he feels in finding out something new.

Given that so much of the Doctor's recent past has been struck from his memory, his interactions with Bill show him putting himself back together. The more she invades his world (invited or otherwise), the more he can put back into the place the person(s) he used to be. It's no coincidence that during his first formal meeting with her in his office we see a selection of his old sonics in a pen pot on his desk. This is a man who used to do things, and no longer does, partly because he needs to look after his vault and partly because he has no reason to do it.

https://www.youtube.com/watch?v=YtjLeSm5O0E&t=7s

And over the course of the following weeks, as their friendship blossoms, Bill reminds the Doctor that he's not as responsible as he claims to be, that he's a nosy person, an interfering person, someone who likes to solve a mystery before anyone else can get to it. He might enjoy teaching science in his inimitable way, but he's far happier let loose, turning up somewhere, digging about until he finds a problem, then solving it and running off again.

Having traveled with Clara for so long that she not only experienced all of his past lives, she became a surrogate for him — whether he remembers it or not — the Doctor now needs a companion who questions everything he does. He needs to restate everything because he's with someone who doesn't know anything about him or his enemies:

https://www.youtube.com/watch?v=YaLficIRInM

Companions are usually the ones to bear the brunt of the consequences of the Doctor's interference, and Bill pays an exceptionally high price for her friendship. But first there's a neat irony in the fact that, as a consequence of seeing the universe through Bill's eyes, the Doctor becomes blinded himself. His sight is eventually restored though, as part of his seemingly infinite supply of second chances, and Bill soon discovers that this doesn't apply to her. As the direct consequence of the Doctor's attempts to rehabilitate Missy, Bill is first shot through the chest, and then converted into a Cyberman.

https://www.youtube.com/watch?v=k4bLMCle9ZY

But amid the tragedy, there's a twist. Bill is rescued by Heather, who offers her the chance to go back to her old life or continue her explorations of space and time in liquid form, and she picks the latter. It's her choice to do this, she leaves behind the defining markers of everything that held her back on Earth — her physical form and social environment — in order to fly among the stars. Her exit from the Doctor's life is the opposite of the tragedy of Donna Noble, and feels oddly life-affirming, because she's been transformed by her time in the TARDIS into a being that represents her best, truest self.

Now go back and read our library of companions from the beginning, with Susan, the Doctor’s granddaughter.