$\sigma_\ell, \; \ell =1,2,3$ are the usual Pauli matrices. The quaternionic imaginary units are identified as, $i = \sqrt{-1}\sigma_1, \;\; j = -\sqrt{-1}\sigma_2, \;\; k = \sqrt{-1}\sigma_3$. Thus, $${\mathbf{q}}= \left(\begin{array}{cc}
x_0 + i x_3 & -x_2 + i x_1 \\
x_2 + i x_1 & x_0 - i x_3
\end{array}\right) \qquad \text{and} \qquad \overline{{\mathbf{q}}} = {\mathbf{q}}^\dag\quad \text{(matrix adjoint)}\; .
 \label{q3}$$ Introducing the polar coordinates: $$\begin{aligned}
x_0 &=& r \cos{\theta}, \\
x_1 &=& r \sin{\theta} \sin{\phi} \cos{\psi}, \\
x_2 &=& r \sin{\theta} \sin{\phi} \sin{\psi}, \\
x_3 &=& r \sin{\theta} \cos{\phi},
 \end{aligned}$$ where $r \in [0,\infty)$, $\theta, \phi \in [0,\pi]$, and $\psi
\in [0,2\pi)$, we may write $${\mathbf{q}}= A(r) e^{i \theta \sigma(\widehat{n})}
 \label{q4},$$ where $$A(r) = r\mathbb \sigma_0 \quad \text{and} \quad
\sigma(\widehat{n}) = \left(\begin{array}{cc}
\cos{\phi} & \sin{\phi} e^{i\psi} \\
\sin{\phi} e^{-i\psi} & -\cos{\phi}
\end{array}\right).
\label{q5}$$ The matrices $A(r)$ and $\sigma(\widehat{n})$ satisfy the conditions, $$A(r) = A(r)^\dagger \quad,\quad \sigma(\widehat{n})^2 = \sigma_0
\quad,\quad \sigma(\widehat{n})^\dagger = \sigma(\widehat{n})
\quad,\quad \lbrack A(r), \sigma(\widehat{n}) \rbrack = 0.
 \label{san1}$$ Note that $\vert{\mathbf{q}}\vert^2  := \overline{{\mathbf{q}}} {\mathbf{q}}= r^2 \sigma_0 = (x_0^2 +  x_1^2 +  x_2^2 +  x_3^2)\mathbb I_2$ defines a real norm on $H$.

Using the above complex representation for quaternions, we defined a set of vector coherent states (VCS) in 