Well, if you have one of the original pressings of the game and you have not patched it, you can return to the toxic caves after the game is over and if you have the item "heart pills" from the Westin murder quest, you can kill yourself in the elevator.
After the usual death scene plays, the screen will stay black without the menu screen opening up. After several minutes, you will begin to hear a hollow echoey brownian noise cave sort of sound. The screen will slowly fade back in to find your character in a pile of that nasty biomass goo that was all around the Master from the first game. Your character will stand up and the usual ambient soundtrack will start playing, but the brown noise will still be there.
Explore this new level, but DO NOT pick any locks. Those areas are off-limits and the developers put some very nasty programming tricks into the code to protect their secrets.
As you continue farther into the level, you will hear the brownian noise continue to increase in volume and the ambient soundtrack you are so used to will exhibit strange artifacts. This may be because of the difficulty of playing two music tracks simultaneously in the Fallout engine which wasn't designed to be able to do this.
Passing locked door after locked door, you will come upon many of the characters you met earlier in the game. Oddly, you only find characters that died, or that would be reasonably be expected to die after you last saw them. Like the official endings, the characters found vary depending on how you played the game...if you were the good guy and tried to solve problems without violence, you will only find a few bad guys and unfortunate victims here. If you slaughtered every town, you will see several hundred characters.
Regardless of what you did earlier, none of the characters will speak to you or react to any action in any way. They can't be pick-pocketed, killed, pushed aside, or healed. If you try to use First Aid, Doctor, or any healing items on them, the game will tell you in the text box, "It's much too late for that, Chosen."
It would appear that this area is purgatory, or possibly hell, based on being populated entirely by dead characters. The final character, standing just before the final door is always a Player Character model from the first Fallout. This is the only character you can interact with and as you approach, the brownian noise reaches a crescendo and the ambient music abrupt cuts out. If you simply bypass him and open the door, the game will play the end credits once again, only with pictures of Hiroshima and Nagasaki victims in the background. This is in extreme poor taste and many have wondered why the creators would be so insensitive. When asked, they deny the sequence exists and call it a hack that was never in the game at all. After it ends, you are greeted with a typical game over screen and are booted to the desktop.
If you talk to the final character, he will explain that he is in fact the Vault Dweller from the original game, your ancestor. He will tell you that he is disappointed in the way you turned out and he will turn his back on you and your character will collapse into a pile of bones in a death animation that I have never seen in the game itself. Afterwards, the game will fade to white and lock up the computer, forcing a hard reset.
There is a third option. Those locked doors I mentioned earlier. This is one door, it's always random as to which door, but if you get lucky in picking you can use any heavy explosive to blow it open. Inside, you will find a single footlocker hold a 10mm pistol with no clip in the picture and no ammo in it. None of your normal 10mm ammo can be used. You can "load" the gun with the "Easter egg" found in the basement of New Reno Arms. Fire this into the head of the final character and the game will cut to an over-the-shoulder video showing a young man playing an unidentified Fallout-like game. Some people claim that this is an early version of either Fallout Tactics or Van Buren, but nothing on the screen seems to fit either of those games. Also neither of those games had started development at the time of the original pressing of Fallout 2. The video itself is poorly lit, with the apparent intention of being a creepy si[her, but nothing spooky actually happens in the video. The man simply plays the unknown game and the video slowly fades into your desktop (a cool trick, I'm not really sure how they managed a gradually translucing screen back then).
Another strange trick is that according to many players, the final character always matches the character they most recently played in Fallout 1. Both gender and their apparel at the end of the previous game are represented. At first this would appear to be a savegame hack, much like Psycho Mantis in Metal Gear Solid, but the trick works even if the original Fallout was played on a different computer without transferring any data to the new PC.
It's recommended that you do not turn on any televisions in your house for several hours after experiencing this extra ending. You will quickly realize that the brownian noise in the game is identical to the noise now coming from your television. Cable, satellite, and even antenna, if you still have that, are all somehow incapable of picking up a signal for some time after the secret ending. All internet connections, however seem to still work. That's how I'm writing this to you now. The funny thing is, I turned my television back off nearly an hour ago, as well as my computer speakers... but I still hear the static getting louder.