  We investigate complex structures on twisted Hilbert spaces, with special
attention paid to the Kalton-Peck $Z_2$ space and to the hyperplane problem. We
consider (nontrivial) twisted Hilbert spaces generated by centralizers obtained
from an interpolation scale of K\"othe function spaces. We show there are
always complex structures on the Hilbert space that cannot be extended to the
twisted Hilbert space. If, however, the scale is formed by rearrangement
invariant K\"othe function spaces then there are complex structures on it that
can be extended to a complex structure of the twisted Hilbert space. Regarding
the hyperplane problem we show that no complex structure on $\ell_2$ can be
extended to a complex structure on an hyperplane of $Z_2$ containing it.
