[Zanth]

More on the fringe of Imogen's style, but certainly similarities in the voice:

Jenny Lewis (and her band Rilo Kiley)
Neko Case (and the New Pornographers in which she sings)
Joanna Newsom
Yuki Chikudate (anything by her really but mostly her band Asobi Seksu)

 

[Amplitude]

Definitely try out Psapp, and maybe Beth Orton's "The Other Side of Daybreak" (Mostly the Four Tet remixes and the last track for Imogen Heap sounding stuff, but the rest is good too)

http://www.youtube.com/watch?v=-OBxL8PiFc8

 

[Sarchi]

How about Dido?

You might also enjoy Laurie Anderson...and maybe some Portishead.

 

[Denver Max]

I second Jenny Lewis/Rilo Kiley, she has a very similar tone to Imogen Heap.