I wouldve loved to see Diana Rigg' s Emma Peel paired with the Third Doctor. I don't think the Doctor could handle all that kick ass competence. I could see her saying to him, "I don't scream Doctor. I'm a professional. "
I always felt like the TV version of Ford Prefect could've been an incarnation of the Doctor, if not another renegade timelord. But #4 or #5 paired with Ford would've been cool to see. The electronic thumb wasn't too different from a vortex manipulator, just without the temporal aspect.
Ford was originally written as basically the Doctor without the restriction of the requirements for the Dctor, as Douglas Adams's reaction from being script editor...
My fantasy Doctor/companion pairings: Jean-Luc Picard and the sixth Doctor. Princess Mary (from The Tudors) and the tenth Doctor.
When I watched Timeless I always pictured Lucy as Barbara, Wyatt as Ian and Rufus as Susan. I can easily see them bickering with the First Doctor. I would pay to see O'Neill from Stargate and Twelve annoying each other. Samantha Carter always showed a gleam in her eyes when she stumbled into new scientific discoveries or real life examples of what she thought was only possible in theory. That sense of wonder she had exploring the universe would fit so well with Ten, it would've been glorious. Walter Bishop from Fringe and the War Doctor guilt tripping together is also something I would've wanted to see.
When Voyager was on-air, I really hoped for a crossover with Doctor Who (I was harbouring notions that some revival with McGann could occur. I really liked the idea of The Doctor meeting The Doctor. And of course, I would've loved to see how the sarcastic, abrasive EMH got on with his rather more gentle (or so he seemed from the TV movie) namesake.
I've said this elsewhere, but-- The 12th Doctor and Bill and Ted. They all travel through time in phone booths and they all play electric guitar. They could have an "ULTIMATE! TIMELINE! TOUR!"
ISTR it was before his episodes came out, but (due to a shorter lead time) after he'd started doing the show.
I've often thought The Doctor would pair up nicely with Scooter from the Go-bots. Mary Poppins is an obvious choice for an adventure or two, but then, I suspect you'd find a tattoo of a snake if you looked at her arm close enough. Maybe Bond teamed up with the third.
Gotta be Moore. Both the third official incarnation and both were dandies. Plus there's an overlap in time (though Pertwee also spans DAF Connery and just misses Lazenby in OHMSS. Yes, obviously the Doctor can time travel but Three did much less than the others.
Or conversely, those characters meeting their doppelgängers. And The Sixth Doctor's in theirs. I would be especially interested in seeing Vila meeting Goudry because they're so different.
McGann's Doctor had such an exuberant way about him in the movie... he might have been able to cheer up Janeway when she was depressed, and there wouldn't have been any of Chakotay's boring "Angry Warrior" crap for her to deal with. A pairing I'd have loved would have been the two elderly Crawley ladies (Violet and Isobel) from Downton Abbey, paired with the Tenth Doctor. After all, Isobel was played by the same actress who played "Harriet Jones, Prime Minister." It would have been easy enough for her to have gone back in time somehow (before her falling out with the Doctor), gotten left behind, and would have had to make a new life for herself. Why should the younger companions have all the fun? And if the Doctor ever dropped in on the Caesars of the 1st-century, it would make perfect sense to say, "For some reason, most of the people in this room look awfully familiar..." Most of the actors/actresses playing the major characters in I, Claudius have also appeared in Doctor Who. The only ones I can think of who didn't were Sian Phillips (Livia) and Patrick Stewart (Sejanus). Not sure about John Castle (Postumus).