Most high level campaigns are Munchkins meet Monty Haul type campaigns were the foulest of players and the foulest of DMs are brought together in one game.
But how do you run legit High Level Characters in a Campaign as the referee and make it work. Here are a few of my thoughts.
1. More problem solving and social interactions
2. Diplomacy
3. The Domain Game is more involved
4. Politics with the royalty and nobility of the realm
5. Quit nerfing your monsters, everything should be Tuckerized from the Kobalds to the Beholders to the Dragons
6. Set hard power caps, so that regardless of level, no one has to borrow a cape from Superman or from Dr Strange
7. Remember test the player, not the character
8. DO NOT fudge die rolls, let the dice fall as they may, read them and weep
9. High Level Characters have friends (maybe) and enemies (definitely)
10. The Bad Guys always cheat