I ship:
First Doctor x Vansell (Thansell) because I think they had Susan's father together.
Master x Rani because they're kind of counterparts.
Ian x Barbara (Barbarian) because they're an iconic pair and they're the original TARDIS pair.
First Doctor x Cameca because she was the Doctor's first on-screen fling.
Steven x Vicki (Sticki) because they might be destined as king Diomedes and Queen Cressida of Argolis.
Dodo x Graham (not the Companion, the guy that worked in the Dodo Chaplet funeral home) because I'm pretty sure they're supposed to be a couple.
Ben x Polly because it's also an iconic ship and they were there for the original Regeneration.
Second Doctor x Jamie because welp. They totally married each other in Season 6B, in a life where he had his 8+ kids with the Doctor, before getting resetted and having them with Kristy instead.
Benton x Yates because they were totally a gay couple.
Sarah x Harry because I think they could have had the chance to become the next Ben x Polly if Ian Marter lived a bit longer.
Harry x Ben because well they're both born in 1942 and it was said that Harry only works on sailors so maybe it could work as an alternative, Harry and Ben are secretly gay for each other, Polly and Sarah were "the beard".
Fourth Doctor x Romana because well they're coworker Time Lords with tons of chemistry.
Adric x Nyssa (Nysric) because I think there was something before Adric died.
Tegan x Nyssa because they had chemistry and were totally space lesbians.
Sixth or Twelfth Doctor x Tegan because they'd have more chemistry than what was seen in the show.
Doctor (Fifth? Sixth? Tenth? Eighth? Twelfth?) x Peri because face it they kinda deserve each other and have chemistry, Nicola Bryant is adorable and peaked at 53 year old and has a bit of a crush on Twelfth\Capaldi and they would have such chemistry, perhaps better than what the Doctor has with River or what Peri once had with the Sixth.
Mel x Ace because well they were the last Classic Who Companions and they met and seemed to have chemistry and would bring the best in each other.
Rose Tyler x Mickey Smith because they were the first couple of the revival, they're like Ian and Barbara genderbent version. If there was Susan around then Rose would have become like Ian.
Martha x Riley Vashtee because they had chemistry together and Riley should really have been a Companion.
Martha Jones x Steven Taylor because she'd see it as a better version of her relationship with Riley (two men with simple, calm personalities, wawy brown hair and grey eyes, that work in red spaceships\space stations), which works over Riley because with Steven being a former Companion it would cause less changes to the timeline, and also because Steven is closer to Martha's time than Riley's, and because they (well, Freema Agyeman and Peter Purves) were hanging out together at the have tea with the Doctors thing and seemed to have chemistry, and actually the relationship could work, the two companions with more calmer and normal personalities and thus more normal-sounding names, they could work off the Tenth Doctor's more dramatic personality.
Martha Jones x Thomas Milligan because they travelled together for an year and he's basically the present-day version of Riley and Steven, since Thomas also has a simple, calm, and heroic personality, and also has wawy brown hair, so basically Martha Jones has a type. She got with Mickey because Tish berated her for getting with Steven since he's not supposed to be born until like 2339 or whatever.
Amy Pond x Rory Williams because well they're also an iconic TARDIS pair and two of the best Companions.
Eleventh Doctor x Rose Tyler because they'd have more chemistry than with Tenth or Ninth.
Nazreen Chaudry x Ace McShane because it kinda works and it ties her with the world of Companions.
Twelfth Doctor x River Song because they're basically married with a 24 year honeymoon. And have so much chemistry.
Clara Oswald x Bill Potts because it's what they deserve. Well technically they have Ashildr and Heather but I think they would have gotten together, if they worked in the same school. And they would pilot the same TARDIS.
Bill Potts x Ace McShane because well they're a lot alike.
Yazmin Khan x Ryan Sinclair because they should've been another inseparable TARDIS couple like Amy\Rory.
Graham O'Brien x Dan Lewis because gay space grandpas.