So I had this idea a while ago of a sort of reality-hopping campaign. Rather than consecutive scenarios taking place in a continuous world, the PCs would regularly slip into different universes (or planes, or dimensions). This would certainly offer aesthetic changes, as well as changes in the availability of technology, magic and other resources; how much it would change the tone and genre I'd have to see.
I see this as something where the PCs would not be in control of their journeys. They might end up shifting inexplicably, either through some kind of affliction, or under the control of higher authorities (be that government agencies or warring deities). Alternatively, it might be some kind of meandering quest: perhaps they're travelling through a series of unmapped portals, trying to get home, or to reach some other destination. Homeward Bounders is probably my main inspiration here, though things like Mighty Max and even The Wishing-Chair share some of the same ideas.
That doesn't necessarily mean the players don't have any control. A DM with fantastic ideas and a great knowledge of what would suit the group might be able to repeatedly conjure up amazing new worlds for them to adventure in, but it's probably best to do some negotiation with the players to see what sort of places they'd like to visit. It's also going to depend how people see the campaign. A serious dimension-hopping storyline, probably with some kind of genuine plot arc going on, facing the gritty reality of adjusting to alien lifeforms and massive swings in technological ability, is going to be a very different kettle of fish from a madcap journey through Robot Dragon World and the Elemental Plane of Sweets.
Switching to different realities would mean different resources become available to the PCs. In some places magic might be non-existent, but in others, you can't get away from the stuff. Some dimensions might offer fabulously advanced technology that makes our most brilliant scientific advances look like the wheel, while others think the wheel is a brilliant scientific advance. I sort of imagine the PCs accumulating motley abilities as they pass through different realities, though not all would always be viable, or indeed safe. You might end up with a solar-power robot arm, four healing spells, a bottled ghost pirate, and expert knowledge of interstellar navigation and flint-knapping. But try explaining the robot arm to a mediaeval mob. Again, a very down-to-earth campaign might tackle the practicalities and problems of augmetics or magic in other dimensions, while a pulpy one might handwave anything that wasn't cool.
To handle something like this, there's basically three options. One is to switch to an appropriate system for each hop, which would mean trying to recreate characters in multiple systems (a time-consuming process, with translation loss and gain every time), getting used to new systems regularly, and working out how adapt items, creatures or abilities appropriately if we want to be able to accumulate cool stuff during the campaign.
Yeah, that sounds like a right barrel of laughs.
A second option would be to pick a single system that people enjoy and are comfortable with, and then basically stick with it regardless. Problems would be hacked or handwaved by the GM. No rules for cybernetic implants in your AD&D rules? Treat it like a magic item. No rules for horrific Lovecraftian magic in your Traveller books? Graft some on, or make a ruling on the fly. You want to stuff Wyle E. Coyote's ghost into a Dalek shell and feed him telepathic instructions to hack the Death Star from the inside, in your Hellcats & Hockeysticks game? ...actually, that one probably works.
On the whole, though, that strikes me as a decent basis for a campaign that happens to veer into unexpected territory, but not a good starting point for one that plans to be esoteric and hotchpotch from the outset. You're just asking for trouble, and rulings you make early on could easily come back to bite you in the posterior.
The third, and I imagine the best, is to start from the outset with an adaptable system that can accommodate a variety of settings, and hopefully genres. Obviously any system is going to suit some things better than others, but something designed for generic gaming will probably do better than a niche game. On the downside, those kinds of systems will lack the depth and richness of more specific systems that focus on particular elements of the game, or on particular tones and genres.
The main generics that I'm aware of are GURPS, BRP and of course, D20. In some ways I'm most tempted by the D20 option, probably because I'm most familiar with it. On the downside, it's quite a fiddly system in some ways, and takes a while to get used to if you aren't. Also, spellcasters are utterly OP after a few levels. On the upside, the fiddliness allows a lot of scope for flavour, genre evocation and creating mechanical differences. It has detailed rules for casting arcane spells, for performing combat manoeuvres, for cybernetic implants and for starship combat. You can temporarily allow or disallow particular classes, spell types, technologies and abilities to reflect character background or dimensional variation. You can vary the background magic level, the abilities granted by different deities. There's a whole array of playable species, many with special feats available reflecting their racial capabilities, like extra-scaly skin or unusually sharp teeth. You can mix classes, allowing characters to gain levels appropriate to the current setting.
On the other hand, a much looser system might be better for a handwavy romp through space and time. In something like BRP, you can just establish some skills and use them for pretty much everything. New and unexpected challenge from a bizarre universe? Invent a new skill. There's simple mechanics for acquiring and improving skills based on use. It's really pretty quick and simple to use. There's bolt-on options for fairies, morris dancing, robotics and superpowers. A downside of the system's simplicity is that signature abilities and tactics are largely non-existent, with differences between characters largely down to skill percentages. This could end up making what should be the cool weirdness of (say) acquiring robot arms or learning from octopus martial artists, into just a few numbers.
It was originally just a silly title for this post, but I've actually got quite tempted by the idea of an Outward Bounders setup. If you're not familiar with Outward Bound, it's one of these outdoor education schemes for young people. They encourage hiking, camping, boating, and that sort of thing. One possibility for a campaign would be to have an organisation that encouraged even more exotic sorts of exploring and independence, sending people off on inter-reality journeys. Portal-hopping could work a bit like orienteering, so they have to collect proof of reaching their various destinations before they can return. If their home culture is sufficiently macho or psychotic, then sending them into quite dangerous journeys would be fine - the same's true to some extent if they're part of some tough organisation.
Coming at it from another angle, maybe the Outward Bounders are - like most kids sent on these things - unwilling participants, perhaps suddenly thrust into adventure by some extradimensional authority that selected them as suitable
victims subjects candidates for this ordeal highly advantageous, character-building leadership programme. Which makes perfect sense to them, but not to the motley band they've imposed this on. Maybe they're all plucked from different worlds (might be fun), with no way home until they complete the programme. Some might relish it, others hate the whole business.
I'm still not sure what kind of tone I'd like - I think you could get about as much mileage out of this by treating it either as a fun romp, a fairly serious fantastical adventure, or a deadly serious straight-faced game of really ridiculous things happening. I mean, the prospect of humourless Dimension Ranger recruits trekking through a planet of Blytonesque talking animals, being press-ganged into a terrifying school and battling their way out of a chocolate prison (and then moving on to a serious Asimovesque robot planet or the Forgotten Realms) really appeals to me.
Anyway, as with most of my ideas I suspect nothing will come of it. But it's a nice thought.