Doctor: You could be so wonderful. You're a genius, you're stone-cold-brilliant, you are, I swear, you really are. But you could be so much more. You could be beautiful. With a mind like that? We could travel the stars, it would be my honour. Because you don't need to own the universe, just see it. To have the privilege of seeing the whole of time and space -- that's ownership enough.
Master: ...I don't know what I'd be without that noise.
Doctor: I wonder what I'd be, without you.
Master: (smiling) Yeah.
He and the Doctor used to be best friends in their schooldays, and have been chasing each other around and aiming ludicrously phallic objects at each other since the 1970s. They're now the last of their kind, making the Doctor even more desperate to help the Master, who is fully aware of what it all sounds like:
The Doctor: I've been alone ever since...but not anymore, don't you see? All we've got is each other!
The Master: Are you asking me out on a date?
Helped along by the sixth Master being specifically cast and acted to resemble the Doctor in his actions and mannerisms. His wife, or as the Doctor indirectly calls her, The Beard, clearly was not willing to let another man run off with her husband though.
The Master was quite clearly being sarcastic during the part of the conversation quoted above. Of far more relevance to this trope is the following telephone exchange.
There's also the Master pushing the Doctor around in his wheelchair, singing along to "I Can't Decide". Although the line doesn't feature in the programme, since Doctor Who is a child-friendly programme, the song contains the line 'F*** and kiss you both at the same time'.
"Lock the doors/And close the blinds/We're going for a ride..."
The Mark of the Rani provided some three-way Foe Yay between the Doctor, the Master and the Rani. Apparently strong enough to deserve a bit of lampshading.
The Doctor: The Rani IS a genius. Shame I can't stand her. [pause] I wonder if I was particularly nice to her she might... no. No, of course not.
The Rani apparently ships Doctor/Master, too, in addition to having Foe Yay with both: You and the Doctor are a well-matched pair of pests. This isn't an isolated comment -- she spends most of the episode making cracks about how she wishes the Doctor and the Master would take their courtship elsewhere and leave her to her experiments.
In the non-canon webcast Scream of the Shalka, the Master has had his consciousness transferred into an android that was (probably) created by the Doctor to save his life. They are now living together semi-peacefully on the TARDIS. Complete with marital bitching and a rather suggestive outgoing message on their answering machine.
Doctor (on recording): *giggle* -- You have reached the good ship TARDIS. *giggle* We're rather -- *giggle* -- busy at the moment, *giggle* *giggle* so leave a message after the beep and we'll try to get back to you before you called. *giggle* -- Stop that! *giggle* *giggle*
The same author wrote the Seventh Doctor book "Human Nature" in which the Doctor recalls "A boy in [his] class who so hated and loved [him] that he kept upsetting [his] experiments." Given Three specifies that they *both* did this, the implications run a little deep.
Then there's the scene with a sobbing Doctor cradling the Master's dying body in his arms in Last of the Time Lords. It's nigh identical to any number of scenes from any number of programs where the hero holds his dying love interest in his arms.
And the Master keeping the Doctor in a dog kennel? Kinky...
The Doctor also seemed very admiring of Professor Yana back when he didn't know who he actually was.
RTD said it himself (spoilers for the 2009 Christmas specials): It's personal for the Doctor. The Master is his enemy, his opposite, and yet so tantalisingly close to being his soul mate. There's something epic about their sheer existence - the last two survivors of an ancient race. It's a clash of the titans. Both of them heading for death, and yet both determined to survive - at any cost!
Russell T Davies: Because he loves him. Honestly, I think he does.
The second part of The End of Time includes the Doctor gushing about how brilliant the Master is and how beautiful he could be if he'd just give up this evil lark. "We could travel the stars; it would be my honour". When the Master shortly thereafter says he doesn't know what he'd be without the drums, the Doctor wonders "what [he'd] be without [the Master]." There's really nowhere else this one can go except to the land of canon.
To which the Master softly replies "Yeah." With John Simm visibly tearing up as he says it.
What with the Doctor saying that the only thing stopping him from running away with the Master into the sunset was the Master's habit of killing people? Not a damn much.
And the Master makes some quite catty comments about the Doctor's fondness for earth girls throughout both parts, and seems unduly pleased with himself when he turns the Doctor's entire favourite species into...himself. Jealous, much?
On a related note, there's the Master cannibalizing the TARDIS into a Paradox Machine, especially since Doctor / TARDIS is now pretty much canon. He really doesn't know how to share, does he?
What about that scene in the Paul McGann film where the Master tackles the Doctor to the ground and tells him:
The Master: You are my life, Doctor.
A history, a book that attempted to put in chronological order all the events in Doctor Who published material up to 2006 made note of the line uttered by the Master as he burned in "Planet of Fire"
Won't you show mercy to your own...[burns]
They note that people think the final word is "brother" but another, less common guess is "husband".
In the interests of balance (i.e. being a spoilsport), it could just as easily have been 'Kind'.
An old favorite from the past: The Sea Devils starts with the Doctor visiting The Alcatraz where he put the Master during their last encounter (While making sure he would be comfortable sincehe was going to be there for quite some time) for no other reason than to make sure he was doing all right - the fact that he happened to stumble on the mystery of the month was a pure coincidence. The Master on his hand plays the role of the model prisoner, makes polite conversation with his guards and claims to have seen the error of his ways. He has a polite conversation with the Doctor and ends it with wishing that the Doctor would visit more often so he can have some intelligent discussions. He then reaches out his hand in friendship. The Doctor reaches for it, changes his mind, and awkwardly waves a goodbye to the Master. Argh!
Four episodes later, when shit has hit the fan and the Master is his usual evil self, he keeps the captured Doctor alive for a convoluted reason. And then when his allies betray him, they end up sharing the same hole together, chummily/sarcastically chatting amongst themselves until rescue arrives.
And then there's The Time Monster, where the Doctor and the Master's TARDISes end up inside each other. Yeah. And then it ends with the Master on his knees, begging the Doctor for mercy.
The whole serial is one huge Doctor/Master slashfest, honestly. It starts with the Doctor having a nightmare about the Master. Wherein he lies on a chaise longue and wakes up to a giant Master looming over him, calling out to the Doctor: "Welcome! Welcome to your new Master!"
The Fifth Doctor serial The King's Demons is made of this. "You have always been my greatest stimulation, my dear Doctor, but now you inspire me."
You have to wonder, too, why the Master went through all the trouble of helping the Fifth Doctor heal from his recent regeneration in Castrovalva before revealing his evil plan.
Because then he couldn't keep feeding the Doctor what basically amounts to a date-rape drug (i.e. valerian).
The date-rape drug comparison isn't inaccurate: the Portreeve's healing tonic includes valerian (sedative, euphoric, attaches to the same receptors as benzodiazepines such as Rohypnol), rosemary (a powerful muscle relaxant) and sanicle (a herb used to staunch... anal bleeding). Honestly.
When the Master returns in Series 8, it's as a female calling herself "Missy." In her introduction, Missy outright refers to the Doctor as her boyfriend. Given that Missy and Twelve dived right into the Friendly Enemy dynamic, a lot of people jumped on the ship.
Missy also loved Twelve's companion Clara Oswald and often seemed to invite Clara to touch her. After the Doctor, Clara's most popular shipping companion is the Master.
This scene neatly summarizes the Doctor and the Master's sexual tension. Yes the subtext is totally intended. According to Sacha Dhawan, the actor who plays the Master, the Master's behaviour in the Series 12 finale was driven by his grief on the Doctor's behalf and was an overblown version of him pulling the Doctor's pigtails.
The War Chief in The War Games is all over the Doctor, with lines such as "We are two of a kind" and "We need each other". This is one reason why some fans believe he's an earlier Master. Noted on the DVD commentary by Frazer Hines:
War Chief:[Having just made sure he's alone with the Doctor] You may have changed your face, but I know who you are.
"Ah", said the Doctor, "˜I see I was standing too close. Invading your personal space. Of course, even from over here I'm invading your personal space. [...] All nestled up under your ribs. Quite intimate, really. Yet we hardly know each other. Love songs have been written about less."
[...] "˜I kept wondering where my heart had got to. [...] Had it joined a club for other lonely ones of its kind? Was it achy? Or breaky? Did it now belong to someone named Daddy?" Sabbath had turned his attention to some papers. The Doctor suddenly stretched out across them, like a cat taking over a computer keyboard. He gazed soulfully into Sabbath's eyes. "˜Shall I call thee Father?"[...]
Sabbath left the room. "I've got you under my skin", the Doctor warbled after him.
In the same book, the Doctor manages to give them a shared Dream Sequence, in which the Doctor turns into a seal and kisses Sabbath. After he turns back, an attack by a giant time squid rocks the boat they're on in this dream sequence, they fall over, and the Doctor ends up on top of Sabbath, panting.
The Doctor gazed up at him limpidly. [...] Sabbath stared at the beautiful, unreadable face.
‘There is that,’ sighed the Doctor. He pulled his feet out of the water and drew his knees up under his chin, wrapping his arms around them, his pale eyes fixed on the foaming falls. There were bits of twig and green leaf in his dishevelled hair. Silva daemonium, thought Sabbath with ironic erudition. To him, at that moment, the Doctor looked much younger than that fool he travelled with. A sick boy. Sabbath wondered idly whether the loss of his heart, which had saved his life, would in the long run kill him.