  We build free, bigraded bidifferential algebra models for the forms on a
complex manifold, with respect to a strong notion of quasi-isomorphism and
compatible with the conjugation symmetry. This answers a question of Sullivan.
The resulting theory naturally accomodates higher operations involving double
primitives. As applications, we obtain various refinements of the homotopy
groups, sensitive to the complex structure. Under a simple connectedness
assumption, one obtains minimal models which are unique up to isomorphism and
allow for explicit computations of the new invariants.
