I’ll give you Jack, but Captain Scarlet there barely qualifies as a full companion xD
The others?
Single female who loses/nearly loses romantic interest, journeys with the Doctor and forms an attachment, romantic inclined for Rose and Martha, not so for Donna, but BFF...each only as a reaction to the one that came before. We are introduced to their family, of each, the Mother is the prime mover, and the father is in some way absent (Martha had a brother, but that largely served to give Reggies something to talk about and give RTD the BBC trifecta...comic, DJ, CBBC presenter...he got one of each in, with Cribbins as the bonus ball.) Wilf is something of an exception to this, though he is also in many ways a continuation of the Mickey theme...a character who stays behind, but is nonetheless involved. All three work in more or less stereotypical female roles...the shop assistant, the caregiver, the ‘secretary’. They are all from working-class backgrounds, with Martha being more middle-class due to her profession (salt of the earth) and all from present day earth, London, the South East. (Ironic, there were probably less London centric stories when the show was made in London.) They are all aged within a rough twenty year span, none particularly old, none particularly young, they are almost deliberately unremarkable, yearning for more than where they have ended up. It’s not impossible to imagine them being acquaintances of each other even.
Now...Amy and Rory deliberately echo all of those set-ups, except, they are seem to be deliberately deconstructing the RTD mold too...the same is very true of Clara too. Moffat basically spends a lot of time stretching the RTD patterns...Rory is Mickey in some ways, in other ways he has everything Mickey lacks (a clear backstory, a career, motivation) Amy is as much the stereotype as her predecessors...right until she isn’t, she has successful careers, does not pine after the Doctor as such, and manages to have a wedding without anyone dying (particularly the groom...except it’s Rory, so he did that earlier. Like Mickey, Rory even got be plastic fantastic, but managed to also be real.) we meet her parents (she manages to be an orphan, then not an orphan, covering Donna and Rose and Martha all at once.) but they are absolutely unimportant and we never ever dwell on them. We meet Rory’s father, but...he is just one of many positive father models in Moffat era Who (they are pretty much all negative stereotypes in the RTD era, even poor old wheeler dealer Pete Tyler) who focuses on a much less dysfunctional family unit than RTD did...largely because he’s not interested in milking it for soap opera drama (every single RTD companion could show up in Albert Square tomorrow and fit right in. Amy and Rory probably wouldn’t...Amy is Scottish, and Take the High Road has been cancelled.) Jackie Tyler is a comedy matriarch, straight out of Only Fools and Horses, Amy is a matriarchal figure in several episodes, but is typically more than that, and more balanced with Rory. It’s a totally different paradigm, before we even get to Clara, the impossible girl who fills several functions but is fundamentally turning into a Doctor character but for deeper reasons than River Song before her. (Someone else has already pointed out her post Pink death wish, so I don’t need to go into that. The only thing that throws me is how she and the Doctor got out of his time stream after Faction Paradoxing it to the nines with Richard E Grant vanishing.) Bill is another character that on the surface shares much with the RTD era...she’s almost a bookend to Rose (bother served chips, both have lost a parent) but was reaching for something by herself...she didn’t need the Doctor to come along and show her how to better herself, she was already doing that, and that’s why the Doctor befriended her.
Now, I have no bone with the RTD era, it had some good stuff, I have no dislike for the companions, but they were not very varied beyond a sort of guess-who game, and didn’t have lives that seemed very deep or varied from each other. Moffat sometimes went off a convoluted deep end (the Pond Family makes the Summers boys in X-Men seem..scratch that, they are about the same level of crazy) but always by going a step further with each character, and hit emotional punches as a natural outgrowth of the stories...not by reaching for the big book of soap opera histrionics.
Going full circle back to Captain Jack, he’s a character that only really gets developed outside of his Who appearances. In Who he’s just a very undeveloped action hero flirt, later a blatant Gerry Anderson in joke (when the Valiant rocks up, it becomes obvious someone’s having a giraffe.) but with a few lines here and there that could have, but never did, go deeper (him being involved in the Time War just never comes up again after his line in Parting of the Ways.) whereas all the Moffat companions went deep and felt exhausted when when we left their wells (apart from Nardole, who is like a comedy introversion of Jack, since he’s functionally immortal, I have a hard time deciding if it’s him or Rory the ever-dying who is more of a commentary on Jack.)
I think it’s telling that even Handles, in one episode, seems to have a bigger arc and impact than poor wasted Martha Jones (she really should have got a better crack at the whip, even if it meant more Torchwood.)
