Given the result in "The Lyapunov dimension of a nowhere differentiable attracting torus" by Kaplan, Mallet-Paret and Yorke, the natural conjecture for the Hausdorff dimension of the graph of $f$ would be $2 - \log(c)/\log(b)$ (take $q(t) = t\,\text{mod}\, 1$ in their introduction).

Given the result in "The Lyapunov dimension of a nowhere differentiable attracting torus" by Kaplan, Mallet-Paret and Yorke, the natural conjecture for the Hausdorff dimension of the graph of $f$ would be $2 - \log(c)/\log(b)$ (take $q(t) = \lfloor t\,\text{mod}\, b\rfloor$ in their introduction).