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For two $n \times n$ complex matrices $A$ and $B$, we define the $q$-deformedcommutator as $[ A, B ]_q := A B - q BA$ for a real parameter $q$. In thispaper, we investigate a generalization of the B\"{o}ttcher-Wenzel inequalitywhich gives the sharp upper bound of the (Frobenius) norm of the commutator. Inour generalisation, we investigate sharp upper bounds on the $q$-deformedcommutator. This generalization can be studied in two different scenarios:firstly bounds for general matrices, and secondly for traceless matrices. Forboth scenarios, partial answers and conjectures are given for positive andnegative $q$. In particular, denoting the Frobenius norm by $||.||_F$, wheneither $A$ or $B$ is normal, we prove the following inequality to be true andsharp: $|| [ A , B ]_q||_F^2 \le \left(1+q^2 \right) ||A||_F^2 ||B||_F^2$ forpositive $q$. Also, we conjecture that the same bound is true for positive $q$when either $A$ or $B$ is traceless. For negative $q$, we conjecture othersharp upper bounds to be true for the generic scenarios and the scenario wheneither of $A$ or $B$ is traceless. All conjectures are supported with numericsand proved for $n=2$.

Bounding the Frobenius norm of a q-deformed commutator