Rare stages:
-random variants with only slimes or something, and higher enemy numbers.
-FIXED heat/crown payout per monster, only based on tier, not depth. it's rare after all.
-probably can't be joined
-(but then you can just join jelly king stages anyway, so why not let people join)

Special stages:
-unlockable by a 3 color combination at stratum creation (even if the stratum isnt completely filled they could be guaranted)
-probably too gimmicky (difficulty of seeing one versus time it takes to develop the stages)

Special material slot:
-grants chance for particular UV type (chance based on mat rarity; type based on who drops it)
-(I'm totally cool with this despite having paid 9k ce for a uv divine avenger)
-grants chance for rare item (requires use of (new) rare material)

Rare Weapons:
-crafting as described
-no UV chance (balance)
-has unique (monster specific) damage behaviour (probably through DMG VS bonus/nerf), charge time, charge type or strike type (no normal swing sheathed sword etc)
-could start out 2*
-can be upgraded up to 5* through use of rare material in special alchemy option.
-it should be rather gimmicky but still worth a slot if you have 4

edit: all details are of course to be worked out in the best way possible and probably not like I described. I just want silly stages every once in a while and weapons that don't follow the typical patterns.

edit: thinking about it, boss reward weapons of new bosses could exhibit such behaviour.

PS: The material related things should be worked out while keeping in mind the price developments after we get the Auction House.
PPS: I hate clockworks tunnels stages still.