This is just a draft of idea I got. It is not finished, detailed or thinked throughout. The idea is to use in a roleplaying game, campaign or even just one adventure. It is not rules specific, so could be used with almost any system.
Parallel Universes And Multiverse
First you might want to check this Wikipedia article to get some idea about this topic. http://en.wikipedia.org/wiki/Multiverse There's lots of to read and I will just pick an idea or two from here and there. But you might want to refer to that Wikipedia article for more detailed article or if you are really interested in this topic, might want to find better articles from Google search.
What I write here, is fictional and doesn't represent my beliefs and should not read as an ultimate truth about life and existence. This is fictional idea for roleplaying purpose, not to be taken seriously from me.
The main idea I will introduce is that every choice you make in your life, splits to another dimension. If you decide not to go to work today but instead take a lazy day, dimensions split in half where in other dimension you went to work. Basically you split in two lives but are only aware of what you live in. As your split dimensional choice in parallel universe does also. You can imagine that there are countless of these multiverses from the beginning of humanity. Several of you, several of other peoples... countless parallel universes.
Also time is relevant. There is no time line in different dimensions and universes. Past is as present as future is.
Accessing Other Parallel Universes
You ever had a dream, where you were in a space ship, or commando mission or even a warrior in barbaric era of fantasy? That actually is reality. At least in some other universe.
Sometimes when you dream, you leave your body and enter a body of one of your parallel self of another multiverse, if also he is sleeping. As there are countless of different versions of you in different kind of realities, it's a big change that some of you is sleeping simultenously. That's when you get to live his life. But the results rarely are permanent as the real person did not live the moments. You just get to live a blink of his life what doesn't affect his life. And vice verca, some other parallel you might get into you when you sleep and live your life in his dream.
But there are some rare exceptions, where you actually live the life of parallel you and the cause is usually amnesia or disorientation of late past.
Also other trained people can access parallel universes via their parallel selves by meditation and astral travel. They might just randomly travel where they get, or try to focus to certain point they knew as their destination. It doesn't always work as the parallel self must be asleep to enter him. Some who master this practice of astral travelling might at will be able to take full control as long as the subject is asleep and make differences and really affect the universe he is.
Usually taking place in parallel universe is just a dream what doesn't affect anything. And usually people don't know they have traveled to other self in another reality as they just think it is an ordinary dream.
Character sees dreams. Maybe when several player characters are asleep they meet at parallel universe together, as they know each other in this reality, they might know each other in another reality also. Or they might not know each other here, but might know each other in one or several of parallel universes. This might be a good story seed. Characters are everyday people not aware of each other in this "real" world, but when they sleep they are together fighting dragons, solving murder mystery in 20's or even in distress in spaceship emergency landed on an isolated planet. When they wake up, they are back in "real" world.
Will they try to seek each other in "real" world to find out what is going on? Will they accidentally see each other at streets and recognize each other from their shared dream? And what does it mean? What will they do, as their strange shared adventure continues every night.
Also one might try to change here and now by trying to travel into past to make his present future different. Problem is to master the arts of astral parallel travelling and also to find right place to make the difference. It will be a campaign to travel into past in different worlds from 20's gangster area to victorian slavery and fantasy world fighting orcs (at some point in history, maybe reality divided in half making orcs and high fantasy happen, who knows). After a change when he wakes up, world might be totally different, and he must travel back in his sleep or astral traveling to fix things and try another route. Results might be, that when he has changed the past and wakes up from his parallel travel world is not 21th century as we know, but dystopia after World War three occured in 70's. Everything can happen!
Game System And Setting To Use
Best effect you get is to use setting of modern times where we do live. It is familiar for players, so all the strange parallel universes of dreams will be more distant and weird. Also setting should be as down to earth as possible, so strange things feel more stranger.
For parallel travel you could use the same setting, but might give parallel universe setting detailed extra skills. You might decide them or let players pick few. It should be quick and it represents the things you do in dreams. Shoot weapons even if you never handled one or wield a sword. That happens in dreams and it is actually residual energy and memories of your parallel self.
For parallel universes you might want to use another game settings of your choice. But you could use the rules same to keep things simple.
This is just a quick idea I wrote down before I forget it. It is not completed, but with a little thinking and tweaking you might quite easily get your own inspiration from this text to run dream based parallel universe adventure.