  Hermitean Clifford analysis is a higher dimensional function theory centered
around the simultaneous null solutions, called Hermitean monogenic functions,
of two Hermitean conjugate complex Dirac operators. As an essential step
towards the construction of an orthogonal basis of Hermitean monogenic
polynomials, in this paper a Cauchy-Kovalevskaya extension theorem is
established for such polynomials. The minimal number of initial polynomials
needed to obtain a unique Hermitean monogenic extension is determined, along
with the compatibility conditions they have to satisfy. The Cauchy-Kovalevskaya
extension principle then allows for a dimensional analysis of the spaces of
spherical Hermitean monogenics, i.e. homogeneous Hermitean monogenic
polynomials. A version of this extension theorem for specific real-analytic
functions is also obtained.
