Witten already discusses deformations for other complexes including the signature complex in his [first paper][1]. He formulated Morse inequalities for the Dolbeault complex on K\"ahler manifolds in a [follow up][2] which seems to have attracted less traction. Wu and some collaborators worked on these, [Mathai-Wu][3], [Wu][4][Wu-Zhang][5],[Wu][6] showing that a Bialynicki-Birula decomposition of the space is needed in the complex setting. In particular this implies Morse inequalities for spin Dirac, signature, self dual, anti self dual, and other twisted complexes in the complex setting.
The technique of deformation had been used earlier by Atiyah to prove results for certain Z_2 graded complexes, and the references in [Zhang's book][7] on the subject covers these.


  [1]: https://inspirehep.net/literature/176416
  [2]: https://www.ias.edu/sites/default/files/sns/files/holomorphic_morse_inequalities-1984.pdf
  [3]: https://arxiv.org/abs/dg-ga/9602007
  [4]: https://arxiv.org/abs/dg-ga/9602008
  [5]: https://arxiv.org/abs/dg-ga/9701009
  [6]: https://arxiv.org/abs/math/9806118
  [7]: https://www.google.com/books/edition/Lectures_On_Chern_weil_Theory_And_Witten/8OfUCgAAQBAJ?hl=en&gbpv=1&dq=Weiping%20Zhang%20book&pg=PR7&printsec=frontcover