  In this paper we will study non-anticommutative perturbative quantum gravity
on spacetime with a complex metric. After analysing the BRST symmetry of this
non-anticommutative perturbative quantum gravity, we will also analyse the
effect of shifting all the fields. We will construct a Lagrangian density which
is invariant under the original BRST transformations and the shift
transformations in the Batalin-Vilkovisky (BV) formalism. Finally, we will show
that the sum of the gauge-fixing term and the ghost term for this shift
symmetry invariant Lagrangian density can be elegantly written down in
superspace with a single Grassmann parameter.
