PLANET OF ORIGIN: Earth
DOCTOR: Tenth
Martha gets a lot of schtick for supposedly rehashing Rose's storyline, being a young woman who falls in love with the Doctor. But Martha is so much more than that. Throughout her time as companion, Martha develops from a lovesick teenager into a woman who can walk away from a relationship that's damaging to her, because there are more important things in life. See, most other companions are forced to cut their time short, either through abandonment by the Doctor, the immutable laws of time and space separating them from the Doctor, or their death. But Martha chooses to stay on Earth where she can look after her family and protect her planet.
PLANET OF ORIGIN: Earth
DOCTOR: Eighth
Before Grace Holloway, any Doctor/Companion romantic tension had to be subtextual. But with Grace, the now half-human Doctor got a lot more kissy. But before Grace got her romance with a Time Lord, she first accidentally killed him on the operating table, trying to save his life. After his regeneration, she takes him into her home and helps him defeat the Master’s plans, although along the way she does get possessed by the Master, helps the Master, and then dies trying to stop the Master. But it’s OK, because she’s brought back to life through timey-wimey business (not the last companion to do so, Rory). But Grace has the shortest tenure of any official companion, given that she refuses the Doctor’s offer to travel around the universe. After all, she's already got a life on Earth (even if she did quit her job).
A female Doctor might be a long way off, but we've got female doctor companions.
But which one would you go for? Let us know below!