Adi Tantimedh at Bleeding Cool imagines a series about the Moment. A sentient weapon that can go anywhere in space and time, and doesn’t want to be fired. Like a TARDIS, only more so.
The article certainly points up the Moment’s importance in The Day of The Doctor, which I hadn’t really considered.
I think a series from its point of view might get a bit samey, but a story could certainly work.
Or a guest appearance as it crosses paths with your characters just when they need it... at that very moment, in fact...