Let $A$ be an $d \times d$ matrix and $(c_1, c_2, \ldots, c_d)$ be a real d-tuple. The $c$-numerical range of $A$ is the set
$$
W_c(A) = \left\{ \sum_{j=1}^d c_j \bra{x_j} A \ket{x_j}: \{\ket{x_i}\}_{i=1}^d \mathrm{\;forms\; an\; orthonormal\; basis\; of\;} \mathbb{C}^d \right\}.
$$
Let $C=\mathrm{\;diag}(c_1, c_2, \ldots, c_d)$.
Then $\mu \in W_c(A) \iff \mu = \tr(CU^\dagger AU)$ for some unitary matrix $U$. This fact motivates to define $C$-numerical range by
$$
W_C(A) = \left\{ \tr(CU^\dagger AU): \mathrm{ \;U \; unitary\; }\right\}
$$
for any square matrix $C$.
For convenience one may define the unitary similarity orbit of matrix $A$ given by the formula
$$
\mathcal{U}(A) = \left\{U^\dagger AU :\mathrm{\;U \; unitary\; }\right\}. $$
Properties of $W_C(A)$ of a matrix $A$ of dimension $d \times d$:
Convexity of $W_c(A)$ and $W_C(A)$ of a matrix $A$ of dimension $d \times d$:
We can generalize the $C$-numerical range $W_C(A)$ to Schatten-class operators i.e. to $C\in\mathcal B^p(\mathcal H)$ and $A\in\mathcal B^q(\mathcal H)$ with condition $1/p + 1/q = 1$, and show that its closure is always star-shaped with respect to the origin [3].
Let $\mathcal{X}, \mathcal{Y}$ denote an arbitrary infinite-dimensional separable complex Hilbert space. Moreover, $\mathcal B(\mathcal X,\mathcal Y)$, $\mathcal K(\mathcal X,\mathcal Y)$ and $\mathcal B^p(\mathcal X,\mathcal Y)$ denote the set of all bounded, compact and $p$-th Schatten-class operators between $\mathcal X$ and $\mathcal Y$, respectively. By $\mathcal B^p( \mathcal{X}, \mathcal{Y})$ we denote all $p$-Schatten-class operators defined by
\begin{equation} \mathcal B^p(\mathcal X,\mathcal Y) := \Big\lbrace C \in\mathcal K(\mathcal X,\mathcal Y)\,\Big|\,\sum\nolimits_{n=1}^\infty s_n(C)^p<\infty\Big\rbrace \end{equation} for $p\in [1,\infty)$ whereas the Schatten-$p$-norm of matrix $A$ \begin{equation} ||A||_p := \Big(\sum_{n=1}^\infty s_n(A)^p\Big)^{1/p} \end{equation} where sequence $(s_n)_{n=1}^{\infty}$ comes from above well-know Schmidt decomposition theorem.
For each $C \in \mathcal K(\mathcal X,\mathcal Y)$, there exists a decreasing null sequence $(s_n(C))_{n\in\mathbb N}$ in $[0,\infty)$ as well as orthonormal systems $(f_n)_{n\in\mathbb N}$ in $\mathcal X$ and $(g_n)_{n\in\mathbb N}$ in $\mathcal Y$ such that \begin{align*} C = \sum_{n=1}^\infty s_n(C)\langle f_n,\cdot\rangle g_n\,, \end{align*} where the series converges in the operator norm.
Moreover, the sequence $(s_n(C))_{n\in\mathbb N}$ is uniquely determined by $C$.
Let $p,q\in [1,\infty]$ be conjugate. Then for $C\in\mathcal B^p(\mathcal H)$ and $A\in\mathcal B^q(\mathcal H)$, we define the \emph{$C$-numerical range} of $T$ to be \begin{equation} W_C (A):=\lbrace \operatorname{tr}(CU^\dagger AU)\,|\,U\in\mathcal B(\mathcal H)\text{ unitary}\rbrace\,. \end{equation}
The properties of $C$-numerical range in infinite-dimensional vector space for Schatten-class operator we can find in [4].