$q$-Witt vectors and $q$-Hodge complexes

In this article, we'll introduce a $q$-variant of Witt vectors and de Rham-Witt complexes. This variant is closely related to the Habiro ring of a number field constructed by Garoufalidis, Scholze, Wheeler, and Zagier, to $q$-Hodge cohomology, and to $\operatorname{THH}(-/\mathrm{ku})$. While most of these connections will only be explored in forthcoming work, the goal of this article is to provide the necessary technical foundation.