I got this idea, after watching SFDebris' review of the episode. And... I liked it! Doctor Who is owned by the BBC.
Doctor Who: The Enemy of the Universe
The Doctor was sitting in his newly re-designed TARDIS and flew around the time vortex, going back and forth from the far future, to the distant past, in order to forget the pain he was feeling. It had been two weeks since Amy and Rory were taken by the Weeping Angels and he was still brooding about it. River had tried to calm him down, but he was unreachable through all his sadness. The TARDIS began to wheeze and the Doctor ran to the console.
"What's wrong, girl? Are you alright?" he asked. The TARDIS responded with a clang, and the Doctor looked at the screen. He saw a man floating in the time vortex. The Doctor frantically pressed buttons and the disheveled man appeared in the control room, and collapsed on the floor. The Doctor noticed the chronometer on the TARDIS console showed the year as 2018. "How did a man get inside the vortex?"
The Doctor walked over to the man, and turned him over, and the Doctor saw a familiar face, the face of his second incarnation. He jumped back, and braced for the time differential to cause a disruption, but it didn't. The Second Doctor sat up and looked at the Doctor.
"Uh, hello. Right, uh, you're the Doctor, right?" The Doctor asked. The man didn't respond, but pointed to his throat. "Oh! Right! Your vocal cords are screwy from the vortex, okay, gimme a second." he whipped out his sonic screwdriver and pointed it at the Second Doctor's throat.
"Where am I?" The Second Doctor asked.
"The TARDIS, your TARDIS, although it's mine now." The Doctor said.
"And who the bloody hell are you?" The Second Doctor asked.
"Your eleventh incarnation. I'm the newest model!" he said smiling.
"Quite right... May I go get changed?" The Second Doctor asked. "And... you've redecorated... I don't like it!"
"Yes! Of course! And, I kept your old outfit after I regenerated from you. It's the second door on the right. It's moved over the past couple of centuries." The Doctor replied. "And this is way cooler than your TARDIS!"
"Right..." The Second Doctor said as he walked out of the room.
"This is odd. Why don't I remember this? And how did I get into the time vortex?" The Doctor asked himself. "This doesn't match up with any event, I never traveled alone as him, I always had Jamie with me!"
A few minutes later, The Second Doctor emerged in his trademark outfit, fiddling with the bowtie and observing the recorder.
"Okay, I need to ask you a few questions, so I know where to return you to." The Doctor said.
"I guess that's alright. Ask away." The Second Doctor said.
"So where are you, Jamie and Zoe, or Dastari and the Androgums? Have the time lords caught you yet?" The Doctor asked.
"I am so sorry, my boy. I don't remember much. I can faintly remember that I was with a scotsman, and a young girl, but I don't remember her name." The Second Doctor said.
"Well then, that narrows it down to after Ben and Polly left. Good. Very good." The Doctor said.
"Ah, good." The Second Doctor said.
"Well then, I'll just run a search through the TARDIS archives until I find the event that you came from." The Doctor said.
"That won't be necessary." The Second Doctor said grabbing the Doctor's arm. "I'll just calibrate it myself."
"Uh, sorry. I respect you, but this is my TARDIS now. It's been upgraded, you'd never be able to work it." The Doctor said.
"I am offended by that statement! I can handle this myself!" The Second Doctor shouted.
"Step back, old man!" The Doctor said. "I will do this!"
"You step back!" The Second Doctor said punching the Doctor in the face. He ran to the console and began to tap random buttons and turning random knobs. He turned back to the Doctor and dragged him into the wardrobe and locked the door.
The Doctor awoke in the wardrobe and grabbed at his pocket, his sonic screwdriver was missing, he ran to the door and began to knock on it, to no avail. He turned back to the wardrobe and started to search through the other outfits, grabbing the frock coat, leather jacket, the cape, and finally settled on pulling out the sonic screwdriver he used in his fourth incarnation.
"Huh, never thought I'd need you again." The Doctor said chuckling as he soniced the door and ran out to the control room. As he ran out he saw the Second Doctor open the TARDIS doors. He quickly soniced the controls and the door closed.
"What the hell?" The Second Doctor asked.
"Why are you doing this?" The Doctor asked. "You're me!"
"No I'm not! I'm not even close to you!" The Second Doctor said.
"Oh my god..." The Doctor's mind flashed with a forgotten memory, a man from Earth, named Salamander, who looked just like his second incarnation. He was stopped when he was thrown into the time vortex. Or so the Doctor had thought. "Salamander!"
"I thought I'd have to keep up this act forever. I can't believe you thought I was that bumbling buffoon! You messed with my vocal cords, giving me his annoying voice! It was easy to trick you!"
"How long has it been? I haven't seen you in centuries." The Doctor said.
"Two months! I was going nuts out there! It was only your idiocy that got me back on your ship!" Salamander said. "And now I've landed back in 2018, and I am going to reclaim my power!"
"That's nice and all, but you are mistaken." The Doctor said.
"What do you mean?" Salamander asked.
"Well, you forgot one thing." The Doctor said.
"What's that?" Salamander asked.
"I have a sonic screwdriver." The Doctor said and aimed it at the console, and Salamander disappeared in a flash of white light. The Doctor walked up to the console and rubbed the console with a handkerchief that had fallen out of Salamander's coat. "Trapped in a time eddy. Temporally orbiting the year 2018, able to be conscious, but trapped for all eternity. That's what he deserves. Evil man! But a handsome devil he was!"
The Doctor fiddled with the console and set the chronometer for Victorian England. He felt like he needed to go far away from where he's been. Too much meddling in the modern era, it was time to go back to a less technologically advanced society. He sat back in his chair and put on his glasses. He glanced at the book sitting on the console, and sighed.
"Even that small adventure wasn't enough for me. I can't do this anymore. I've retired." The Doctor said and sulked back in his chair. "Stopped the enemy of the world, and what thanks do I get? None!"