In his famous work, Supersymmetry and Morse theory, Witten perturbs de Rham complex by perturbing the exterior derivative
$$d_h=e^{-ht}de^{ht}.$$
And he proves Morse inequality using some spectral information of the Laplacians of the complex.
Has similar perturbation on the the other elliptic complexes been studied? Complexes like Dolbeault complex or signature complex, Or maybe in general, twisted spin complexes. Can one expect new information from such a perturbed complexes?