I wouldn't call it quasi-companion, but yeah, that's from the old series. The Master has always been an evil maniac who would be willing to destroy an entire planet, but he has always been portrayed as an old friend of the Doctor, who he looks up to and admires a lot. Most of his plans were very much aimed at destroying the Doctor, but whenever they got in a room together they would start talking like old university mates. Very much the vibe you'd get in old Bond movies whenever he would sit at a dinner table with a villain.
Usually the Master would have some insane elaborate scheme with some other alien race, who would turn against him and then he'd have to work with the Doctor to sort out his own mess. There have definitely been a lot of scenes of them working together.
In fact, even though his schemes were often far too elaborate and he'd be far more successful if he would just make a more straightforward plan, they would often work okay for him, and he would only start fucking it up when he realizes the Doctor has shown up. He pretty much abandons his entire plan at that point to focus purely on getting the Doctor. The obsession is real with that guy. Moffat used to joke that the Master's secret history with the Doctor (which has never been confirmed, just hinted at) was that the Master was obviously in love with the Doctor, but that the Master was the kind of guy who would not only never consider himself as being gay, the very fact that that was an option would never enter his head. He would just be totally oblivious to what his feelings for the Doctor meant, he just had this inexplicable lifelong desire to find and blow up the Doctor, which he could have done a thousand times if he would just stop sweet talking him whenever he finally finds him and instead just shoots him in the face.
From the moment I read that on, the interactions between Delgado and Pertwee have only become more beautiful. It just fits so perfectly. The theory mostly just fits those two incarnations of the Master and the Doctor though - it's not a direction I want the modern series to take. But as an unspoken undertone of the Third Doctor era it's just perfect.
I also think that (awesome) Missy speech about her friendship with the Doctor was a deliberate reference to that, as the unofficial Moffat-Master theory is pretty well known in fan circles, and doing that with a female Master and male Doctor would change the entire tone of the series. It's good that he did that.
