We assume that the manifold with boundary, X, has a Spinℂ-structure with spinor bundle S/. Along the boundary, this structure agrees with the structure defined by an infinite order, integrable, almost complex structure and the metric is Kähler. In this case the Spinℂ-Dirac operator ð agrees with ∂¯b+∂¯∗b along the boundary. The induced CR-structure on bX is integrable and either strictly pseudoconvex or strictly pseudoconcave. We assume that E→X is a complex vector bundle, which has an infinite order, integrable, complex structure along bX, compatible with that defined along bX. In this paper we use boundary layer methods to prove subelliptic estimates for the twisted Spinℂ-Dirac operator acting on sections on S/⊗E. We use boundary conditions that are modifications of the classical ∂¯-Neumann condition. These results are proved by using the extended Heisenberg calculus.