Since Doctor Who returned in 2005, several companions have traveled aboard the TARDIS and, unfortunately, they haven't all been treated equally by The Doctor. While the Timelord has formed many strong romantic connections with some companions, like Rose, and become best friends with others, like Donna, this isn't always the case.
Sometimes, The Doctor fails to realize how special their companions are, only appreciating their worth when it's too late (like with Martha). It is something that has frustrated the viewers for years since it's likely that this newfound discovery could have changed the pair's dynamic forever and made them a force to be reckoned with. Hopefully, in the near future, this is something the writers would rectify and fix so that no one is underappreciated ever again.
Graham O'Brien
Graham joined the TARDIS in season 11 of the show following the death of his wife. With three companions in the TARDIS at this time, it's difficult for The Doctor to form a close bond with any of them individually.
Unfortunately, while he has a lot of respect for the Timelord, affectionately referring to them as 'Doc,' he doesn't get the same level of support as other companions have done in the past (with a prime example being when Graham confides in The Doctor about his fears over his cancer returning but is brushed off). While the moment is intended to highlight the differences in this particular incarnation of the Timelord, Graham deserves more. He shows himself to be capable and supportive on a number of occasions so it's disappointing to see him overlooked in this way by someone he admires so much.
Captain Jack Harkness
Jack first appears in the first season of New Who, becoming a companion to the Ninth Doctor and he even got his own spin-off, Torchwood. Despite some initial bickering between him and The Doctor, the latter did eventually accept him as a companion. That being said, The Doctor abandons Jack after he is revived by Rose rather than helping him adjust to his immortality.
This abandonment feels particularly cruel, especially considering The Doctor's other companion, Rose is actually responsible for Jack's immortality and doesn't receive anything similar to the same treatment. When The Tenth Doctor is reunited with Jack, he does warm up to him again; however, he's never appreciated as much as other companions. Although Jack has done many shameless things, he deserves more recognition from The Doctor, especially considering he defends the Earth when they're not around.
Rory Williams
In season 5 of the show, Rory's girlfriend, Amy, becomes The Doctor's companion and he later joins them in the TARDIS. Although there's a lot of jealousy on Rory's end when he first meets The Doctor, he does later accept him as a friend. That being said, The Doctor still overlooks Rory several times in favor of Amy and puts his safety at risk as a result.
Bizarrely, Rory also becomes The Doctor's father-in-law which sets him apart from other companions. Even when they are taken by the Weeping Angels, The Doctor's focus mainly seems to be on the loss of Amy, even though Rory has also been traveling with them for a long time at this point. While he is definitely just as capable as Amy, Rory often gets overlooked by the Time Lord.
Nardole
Nardole is one of The Doctor's companions in season 10 of the show and plays a key role in some of the best episodes from the Steven Moffat era. Despite being with The Doctor for a long period, Nardole always feels secondary to Bill. The show mainly focuses on the relationship between Twelve and Bill, with Nardole mostly being used for comic relief.
The back and forth banter between Nardole and The Doctor is an important part of their relationship and it's refreshing to see a different kind of companion dynamic. When The Doctor first goes off traveling with Bill, Nardole stays behind in order to guard Missy in the vault which shows he's trusted; however, this is never fully appreciated. Ultimately, Nardole is often the punchline of jokes in the show and The Doctor rarely takes him seriously either.
Yasmin Khan
In the latest episode of the show, The Doctor finally acknowledges Yaz's feelings and finally realizes how much she does for her. Previously to this, The Doctor often snapped at Yaz and shunned her for asking too many questions. While the character isn't given much screen time in season 11, she becomes more independent and rebellious in season 12.
Despite being one of the thirteenth Doctor's best companions, she often falls into the background when Graham and Ryan are also traveling in the TARDIS. After they leave, Yaz does become more like the traditional primary companion but she is constantly shut down by The Doctor in season 13. Although this may be them trying to keep Yaz at arm's length as they've become aware of her feelings, the treatment does feel unfair and unwarranted at times, especially considering the companion's admiration of The Doctor.
Dan Lewis
Dan joins the Doctor and Yaz in season 13 but with so much going on here, his development is limited. Some fans were worried adding another companion into the mix would take away screen time from the blossoming romance between The Doctor and Yaz. That being said, Dan is the first person to notice the spark between the pair and encourages them to communicate with each other.
Although the character hasn't appeared in many episodes, he has formed a strong friendship with Yaz which has been lacking previously within the current era. However, as Dan spent years trapped in the past and hasn't actually spent much time with The Doctor, the dynamic of their relationship is still fairly unclear. If he'd gotten more time in the show, Dan might have become more appreciated by The Doctor and transitioned from Yaz's friend to a proper companion.
Martha Jones
Martha is arguably the companion most underappreciated by The Doctor, especially considering how much she actually does for them. Some fans even believe that Martha and The Doctor weren't real friends as the latter is still hung up on Rose during their travels together. Unlike with many other companions, The Doctor takes several episodes to invite Martha on board the TARDIS full-time.
Fairly early on into their travels, it becomes clear that Martha has romantic feelings for The Doctor which eventually forces her to leave the TARDIS, something which could certainly be repeated when Yaz departs. Martha helps to hide The Doctor when they are forced to become human and she walks the Earth in order to save everyone from The Master. Despite all these incredible things she does, The Doctor still fails to recognize her brilliance, constantly comparing her to Rose, which is only reiterated by her brave decision to walk away.