We investigate complex structures on twisted Hilbert spaces, with special attention paid to the Kalton-Peck Z2 space and to the hyperplane problem. We consider (nontrivial) twisted Hilbert spaces generated by centralizers obtained from an interpolation scale of Köthe 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 Köthe 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 ℓ2 can be extended to a complex structure on an hyperplane of Z2 containing it.

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