Neutral Milk Hotel - completely different from the Lips, but a similar kind of sad, hopeful, triumphant set of songs that the Flaming Lips specialize in. The Aeroplane Over the Sea is one of the best albums of all time.
The Music Tapes - One of the members of Neutral Milk Hotel made this collection of warbling, fragile, high-pitched, experimental extreme lo-fi with dozens and dozens of exotic and homemade instruments. Might appeal to Lips fans.
All the other Flaming Lips albums - They started out more primitive psychedelic punk and only evolved into their lush, superproduced form with Zaireeka. But Zaireeka, The Soft Bulletin, and Yoshimi Battles the Pink Robots are of a kind (especially the first two, which were both recorded at the same time). I actually like some of their earlier work, like A Priest Driven Ambulance and Clouds Taste Metallic, a little more than I like the Soft Bulletin.
Galaxie 500 - Quirky, soaring, transcendental indie pop.
The Robot Ate Me - Much more modest in scale than the above-mentioned bands, but good.
Islands - Islands are the new version of The Unicorns, with most of the original members. If you like The Unicorns, you'll like them.
Lettuce Prey - Many of his songs are on his website, ltprey.com. Check them out.
Penelope Brain - An Australian who seems to be deliberately emulating the Soft Bulletin, down to songs about sad robots. Pretty good.
http://music.download.com/penelopebrain/3600-8618_32-100080660.htmlDaniel Johnston - Has been making crazy, off-key but tuneful music longer than any of these other guys. He's for hard-core lovers of outsider music only, though; his lisping voice can get on some people's nerves.