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The topological Hochschild homology of a ring (or ring spectrum) R is an S1-spectrum, and the fixed points of THH(R) for subgroups C_n of S1 have been widely studied due to their use in algebraic K-theory computations. Hesselholt and Madsen proved that the fixed points of topological Hochschild homology are closely related to Witt vectors. Further, they defined the notion of a Witt complex, and showed that it captures the algebraic structure of the homotopy groups of the fixed points of THH. Recent work defines a theory of twisted topological Hochschild homology for equivariant rings (or ring spectra) that builds upon Hill, Hopkins and Ravenel's work on equivariant norms. In this paper, we study the algebraic structure of the equivariant homotopy groups of twisted THH. In particular, drawing on the definition of equivariant Witt vectors by Blumberg, Gerhardt, Hill and Lawson, we define an equivariant Witt complex and prove that the equivariant homotopy of twisted THH has this structure. Our definition of equivariant Witt complexes contributes to a growing body of research in the subject of equivariant algebra.