  We give a new proof that the sphere S^6 does not admit an integrable
orthogonal complex structure, as in \cite{LeBrun}, following the methods from
twistor theory.
  We present the twistor space of a pseudo-sphere
S^{2n}_{2q}=SO_{2p+1,2q}/SO_{2p,2q} as a pseudo-K\"ahler symmetric space. We
then consider orthogonal complex structures on the pseudo-sphere, only to prove
such a structure cannot exist.
