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?
Hormander's approach to solving the $\bar \partial$ problem is basically this, and his paper is from 1965, predating Witten's work by a couple of decades! By varying the "weight" function $h$, you can get families of estimates on the solution of $\bar \partial$ problem. Check out Hormander's 1965 ACTA paper for more details. Really a fabulous paper. He does not make explicit that he is "perturbing the $\bar \partial$-complex", but that is exactly what he is doing. I am sure Witten was reading Hormander.
It is also possible to perturb this complex in more dramatic ways. My advisor (Jeff McNeal) has done a lot of this work. You might be interested in his survey paper "$L^2$ Estimates on Twisted Cauchy Riemann Complexes".
My thesis work is basically applying such twisted complexes (which can sometimes give estimates which are unavailable without the twisting) to prove some new results on approximation by holomorphic functions in $\mathbb{C}^n$.
Witten already discusses deformations for other complexes including the signature complex in his first paper. He formulated Morse inequalities for the Dolbeault complex on K"ahler manifolds in a follow up which seems to have attracted less traction. Wu and some collaborators worked on these, Mathai-Wu, Wu, Wu-Zhang,Wu 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 on the subject covers these.
Demailly's (asymptotic) holomorphic Morse inequalities (quoted in an earlier comment), inspired by Witten's approach, are different from Witten's (equivariant) holomorphic Morse inequalities, and have been a subject of continued study. Hidden in Witten's approach is an idea of rescaling the metrics of (flat) vector bundles using Morse functions, whence the gradient of the Morse function can be thought of as a curvature term (appearing in the Bochner formula which is key to the deformation arguments), whereas Demailly's approach is to take tensor powers of line bundles whence the curvature changes directly. The curvature is what is being deformed.
Dan Popovici deforms the exterior derivative operator on a complex manifold to $D_{\eta}=\eta \partial+\bar\partial$ for a positive function $\eta$. He was apparently led to this construction by Witten's work. He proves a collection of vanishing theorems on noncompact complex manifolds depending on properties of the function $\eta$.
I'm not sure if this is what you're looking for, but in http://arxiv.org/abs/math/0108185 Dunkl and Opdam define a deformed differential $d(k) = d + \Omega(k)$ on the polynomial de Rham complex on a vector space with an action by a complex reflection group $G$. Here $k$ is a parameter, $d$ is the standard de Rham differential, and $\Omega(k)$ is a 1-form depending on the parameter $k$. The (nontrivial) proof (Thm 2.9) that $d(k)$ is actually a differential (i.e. that it squares to 0) is used to show that the Dunkl operators associated to the complex reflection group $G$ commute. Proving this commutativity directly would require difficult computations.
(The Dunkl operator $T_\epsilon(k)$ for $\epsilon \in V$ is a deformation of the directional derivative $\partial_\epsilon$ depending on the parameter $k$, see formula (5). Dunkl defined these operators for real reflection groups in 1989, which is why they have that name.)
