  The theory of slice regular functions of a quaternion variable is applied to
the study of orthogonal complex structures on domains \Omega\ of R^4. When
\Omega\ is a symmetric slice domain, the twistor transform of such a function
is a holomorphic curve in the Klein quadric. The case in which \Omega\ is the
complement of a parabola is studied in detail and described by a rational
quartic surface in the twistor space CP^3.
