Let $A$ be an $N \times N$ matrix. The q-numerical range of $A$ is the set: \[ W_q(A) = \left\{ z \in \mathbb{C}: z = \bra{x}A\ket{y}, |\braket{x}{y}| = q, \ket{x} \in \mathbb{C}^N, \ket{y} \in \mathbb{C}^N, \braket{x}{x} = 1, \braket{y}{y} = 1 \right\}. \] Note that in the case $q=1$ we get back the standard numerical range.