Let $\mathbb{H}$ be the set of all elements, called quaternions, of the form $q = q_{0}+q_{1}i+q_{2}j+q_{3}k$, where $i,j,k$ are quaternion units satisfying: \begin{equation}\label{Equation:multiplication} i\cdot j = -j \cdot i, \; \; j\cdot k = -k \cdot j,\; \; k\cdot i = -i \cdot k \; \; \text{and}\;\; i\cdot j \cdot k = -1. \end{equation} For a given $q \in \mathbb{H}$, we define the real part, re$(q):= q_{0}$ and the imaginary part, im$(q):= q_{1}i+q_{2}j+q_{3}k$. The conjugate and the modulus of $q$ respectively given by \begin{equation*} \overline{q}= q_{0} - ( q_{1}i+q_{2}j+q_{3}k)\; , \; |q|= \sqrt{q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2}}. \end{equation*}

Let us denote the class of all $n\times n $ matrices over $\mathbb{C}$ and $\mathbb{H}$ by $M_{n}(\mathbb{C})$ and $M_{n}(\mathbb{H})$ respectively.

Let $A = [\;q_{rs}\;]_{n\times n}\in M_{n}(\mathbb{H})$. Since $q_{rs} = a_{rs} + b_{rs}\cdot j$ for some $a_{rs}, b_{rs} \in \mathbb{C}$, then $A_{1}:= [\; a_{rs}\; ]_{n\times n},\; A_{2}:= [\; b_{rs}\; ]_{n\times n}\in M_{n}(\mathbb{C})$ and $ A=A_{1}+A_{2}\cdot j$. Define \begin{equation*} \chi_{A}:= \begin{bmatrix} A_{1} & A_{2}\\-\overline{A}_{2}& \overline{A}_{1} \end{bmatrix}_{2n\times 2n} \in M_{2n}(\mathbb{C}), \end{equation*} then the map $\xi \colon M_{n}(\mathbb{H})\to M_{2n}(\mathbb{C})$ defined by $ \xi(A) = \chi_{A}, \; \text{for all}\; A \in M_{n}(\mathbb{H})$ is an injective real algebra homomorphism. It is clear that $\|A\| = \|\chi_{A}\|$, where $\|\cdot \|$ denotes operator norm in the respective algebra.

Let $A \in M_{n}(\mathbb{H})$. Then:

- The quaternionic numerical range of $A$, denoted by $W_{\mathbb{H}}(A)$, defined as \begin{equation*} W_{\mathbb{H}}(A):= \big\{\langle X, AX \rangle_{\mathbb{H}}: X\in S_{\mathbb{H}^{n}} \big\}, \end{equation*} where $S_{\mathbb{H}^{n}}: = \big\{X \in \mathbb{H}^{n}: \|X\|=1\big\}$.

- The quaternionic numerical radius of $A$, denoted by ${\mathop{w}}_{\mathbb{H}}(A)$, defined as \begin{equation*} {\mathop{w}}_{\mathbb{H}}(A):= \sup\big\{|q|: q \in W_{\mathbb{H}}(A)\big\}. \end{equation*}

- For each slice $\mathbb{C}_{m}$ of $\mathbb{H} \; (m \in \mathbb{S})$, we call $W_{\mathbb{H}}(A)\cap \mathbb{C}_{m}^{+}$ as a $\mathbb{C}_{m}$- section of $W_{\mathbb{H}}(A)$. In particular, we denote the complex section of $W_{\mathbb{H}}(A)$ by $W_{\mathbb{H}}^{+}(A)$ i.e., \begin{equation*} W_{\mathbb{H}}^{+}(A):= W_{\mathbb{H}}(A)\cap \mathbb{C}^{+} , \end{equation*} where $\mathbb{C}^{+}= \{\alpha + i \beta :\; \alpha \in \mathbb{R},\; \beta \geq 0\}$.

Let $A \in M_{n}(\mathbb{H})$. Then the projection of $W_{\mathbb{H}}(A)$ onto the complex plane is denoted by $W_{\mathbb{H}}(A :\mathbb{C})$ and it is defined by \begin{equation*} W_{\mathbb{H}}(A :\mathbb{C}) = \{co(q); \; q \in W_{\mathbb{H}}(A)\}, \end{equation*} where $co(q) = q_{0}+q_{1}i$, for $q = q_{0}+q_{1}i+q_{2}j+q_{3}k \in \mathbb{H}$.

Let $A\in M_{n}(\mathbb{H})$. Then $W_{\mathbb{H}}(A:\mathbb{C}) = W_{\mathbb{C}}(\chi_{A})$. Moreover, if $W_{\mathbb{C}}(\chi_{A})$ is convex by Toeplitz-Housdroff theorem, then $W_{\mathbb{H}}(A:\mathbb{C})$ is convex. In particular, for a self-adjoint matrix $A \in M_{n}(\mathbb{H})$, \begin{equation*} W_{\mathbb{H}}(A) = W_{\mathbb{H}}(A:\mathbb{C}) = W_{\mathbb{C}}(\chi_{A}) \end{equation*} is a convex subset of $\mathbb{R}$.

Let $A \in M_{n}(\mathbb{H})$. The following properties hold true:

- $W_{\mathbb{H}}(A)$ is compact in $\mathbb{H}$

- $W_{\mathbb{H}}(\alpha I + \beta A) = \alpha + \beta W_{\mathbb{H}}(A)$, for every $\alpha, \beta \in \mathbb{R}$.

- If $B\in M_{n}(\mathbb{H})$, then $W_{\mathbb{H}}(A+B) \subseteq W_{\mathbb{H}}(A)+W_{\mathbb{H}}(B)$.

- $W_{\mathbb{H}}(U^{\ast}AU) = W_{\mathbb{H}}(A)$, for every unitary $U \in M_{n}(\mathbb{H})$.

- $W_{\mathbb{H}}(A^{\ast}) = W_{\mathbb{H}}(A)$.

In general quaternionic numerical range of matrices over the ring of quaternions is not necessarily convex. For example, \begin{equation*} A = \begin{bmatrix} k &0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}_{3\times 3} \in M_{3}(\mathbb{H}) \end{equation*} has $k, -k \in W_{\mathbb{H}}(A)$, but $0 = \frac{k}{2} - \frac{k}{2} \notin W_{\mathbb{H}}(A)$. To see this, assume that there is a $X: = \begin{bmatrix} x&y&z \end{bmatrix}^{T} \in S_{\mathbb{H}^{3}}$ such that \begin{equation*} 0 = \big\langle X , A X \big\rangle_{\mathbb{H}} = \overline{x}kx+ |y|^{2}+|z|^{2} \end{equation*} i.e., $|y|^{2}+|z|^{2} = -\overline{x}kx $. This is contradiction since $\overline{\overline{x}kx} = - \overline{x}kx $ and $|y|^{2}+|z|^{2}$ is real. It shows that $W_{\mathbb{H}}(A)$ is not convex.

Now we provide some additional equivalent conditions for the convexity of quaternionic numerical range.

Let $A \in M_{n}(\mathbb{H})$. Then the following are equivalent [1]:

- $W_{\mathbb{H}}(A)$ is convex.

- $W_{\mathbb{H}}(A:\mathbb{C}) = W_{\mathbb{H}}(A)\cap \mathbb{C}$.

- For every $X \in S_{\mathbb{H}^{n}}$, there exists a $Y\in S_{\mathbb{H}^{n}}$ such that \begin{equation*} 2\; |\text{im} (\langle Y, AY\rangle_{\mathbb{H}})| = |\langle X, AX\rangle_{\mathbb{H}}\; i - i\; \overline{\langle X, AX\rangle}_{\mathbb{H}}|. \end{equation*}

Let $A=D+N \in M_n(\mathbb{H})$, with $D$ a diagonal matrix with real entries, $N$ nilpotent and cycle-free matrix. Then, $W_{\mathbb{H}}(A)$ is convex. [2]

More theorems regarding the convexity of quaternionic numerical range we can see in [3].