Absolutely correct, BG.
Turanil: There are LOTS of places in which being an epic character makes sense. PCs played up to 20th level should become the movers and shakers of their homeworlds, mighty forces on par with individuals like the Chosen of Mystra and the Red Wizard Zulkirs in the Forgotten Realms. At this point, adventures can center around several things:
1) Diplomacy and intrigue. All the power of a rampaging barbarian or a mighty wizard can mean next to nothing in the game of thrones. Epic characters probably are already the rulers of their own realms or perhaps the powers behind the throne; as their personal power hits an apex, it stands to reason that more adventures will center around roleplaying, where while epic powers will prove terrifically useful, they'll need to be wielded in a manner more subtle than called for in a typical dungeon crawl.
2) Protecting the world from Things from Beyond. One of the great tropes of some fantasy-related genres is that the universe is far larger than our primitive understanding lets us believe. The things that might inhabit the realms beyond our own are incomprehensible and potentially deadly. The task of epic characters might well be to protect their beloved world from these outside threats; a lonely battle unknown to any of the world's other inhabitants, and one that's fought in places and on a scale completely unimaginable to them.
3) Exploring the multiverse. As characters reach epic levels, they may get bored of their homeworld and wish to explore or even found realms in the planes beyond; what with the genesis spell, this is within the power of wizards even at (high) nonepic levels. Exploring other planes and founding extraplanar realms means having to deal with creatures on those planes or with potential usurpers, all of whom are likely to be terrifically powerful and worthy epic challenges.
4) Uncovering hidden threats. Certain worlds (like the FR) have long-buried threats held within them; the phaerimm, the evil beings inhabiting Myth Drannor, the Twisted Rune. These threats can conceivably exist below the radar of adventurers for their whole careers, only to be uncovered (and subsequently battled) once the adventurers hit epic level.
What are some key elements to play up in an epic campaign?
1) Setting. This is the big one. "Weird alternate planes," as you said; cloud castles; mighty Underdark realms shrouded in darkness; hidden castles at the bottom of the sea; elemental strongholds floating on the lava of active volcanic craters; lonely dead cities floating in the void of the Negative Energy Plane. These settings are immensely colorful, plausibly qualify to be home to beings capable of challenging epic characters, and may require epic powers just to survive.
2) Weird magic, some of it controllable by the characters. At these levels, extreme effects (entire dead magic planes, eternal whirlwind storms that spit out elder air elementals), are quite plausible and within the powers of the characters to combat. Again, go all out!
3) Pacing: I've always said that epic-level D&D campaigns must rely more on Justice League of America and less on Lord of the Rings in how action is paced. While LotR may form the archetype of the epic fantasy tone, the action in epic-level games is of the superhero style. Sessions should move rapidly in a "cut" style from scene to scene due to the ubiquity of teleportation magic and the rapid pace of combat. Games will jump from plane to plane and halfway across the world in the blink of an eye.
As to the mechanical issues: Well, it's about judicious DM-ing. One immediate thought:
1) Control epic spellcasting. Plainly, epic ability buffs should obey a quadratic rather than linear progression, just like items do. You want to use the fortify seed to bump up ability scores? Fine. It's Spellcraft DC 26 + bonus squared. So a +1 increase is still DC 27, a +2 increase is DC 30, a +3 increase is DC 35, etc. This yields quite reasonable epic DCs anyway; A spell that grants a +8 bonus to an ability score is DC 90, which is relatively easy to match with mitigating factors (DC 70 with a casting time of 10 minutes, or DC 59 if your bard buddy throws in a 6th-level spell). Compare with an epic item, and you'll see that the cost is quite competitive (the spell costs between 531,000 and 630,000 gp to develop, can be cast on multiple targets, and is slotless; the item costs 640,000 gp).