  Introducing a non-compact version of the Hopf map, we demonstrate remarkable
close relations between quantum Hall effect and twistor theory. We first
construct quantum Hall effect on a hyperboloid based on the noncompact 2nd Hopf
map of split-quaternions. We analyze a hyperbolic one-particle mechanics, and
explore many-body problem, where a many-body groundstate wavefunction and
membrane-like excitations are derived explicitly. In the lowest Landau level,
the symmetry is enhanced from $SO(3, 2)$ to the $SU(2, 2)$ conformal symmetry.
We point out that the quantum Hall effect naturally realizes the philosophy of
twistor theory. In particular, the emergence mechanism of fuzzy space-time is
discussed somehow in detail.
