Purely FWIW, I'd love for the Doctor to have a Companion that was totally anti-Doctor. Someone that he has decided to keep around because:
- They really are brilliant or just have a lot of potential and he believes that they deserve more than the life they currently lead;
- The Doctor is more than a little fascinated with them, despite himself (the last of their culture or something);
- They're so fantastically dangerous that he really prefers to have them somewhere that he can keep an eye on them 24/7.
As an example? Look up Douglas Adams' original treatment of 'Doctor Who and the Krikkitmen'. They pretty much anticipate Terminators about 10 years before the James Cameron film - human-looking combat robots but that are
true robots rather than androids ('synthetic men').
(1) One idea I had for this never-completed mini-arc was for him to fiddle with the CPU on one of the Krikkitmen using his screwdriver to get its co-operation. Much to his shock, it slowly becomes fully sentient through the length of the arc. His morality prevents him from just destroying it at the adventure's end, so he brings it with him as a Companion, hoping to teach it a method of conflict resolution that doesn't summarise as 'Point gun at problem; pull trigger until problem is no longer there'.
Originally, this story was a Fourth Doctor story so Sarah Jane Smith was the default Companion. I guess I just liked the idea of having someone other than the Doctor on-hand to rescue her whenever she walked into the Monster of the Week.
FOOTNOTE
(1) - In classic sci-fi, androids tend to be more like Blade Runner Replicants than anything else - more biological than mechanical. True robots are those with non-organic 'brains', no matter how many biological components they have.