At the level of formulas, the three quaternionic units $i_a$, $a\in~\{1,2,3\}$, in $\mathbb{H}\cong \mathbb{R}^4$ satisfy
$$i_a i_b ~=~ -\delta_{ab} + \sum_{c=1}^3\varepsilon_{abc} i_c, \qquad\qquad a,b~\in~\{1,2,3\}, \tag{1}$$
while the three Pauli matrices $\sigma_a \in {\rm Mat}_{2\times 2}(\mathbb{C})$, $a\in~\{1,2,3\}$, $\mathbb{C}=\mathbb{R}+\mathrm{i}\mathbb{R}$, satisfy
$$\sigma_a \sigma_b ~=~ \delta_{ab} {\bf 1}_{2\times 2} + \mathrm{i}\sum_{c=1}^3\varepsilon_{abc} \sigma_c\quad\Leftrightarrow \quad \sigma_{4-a} \sigma_{4-b} ~=~ \delta_{ab} {\bf 1}_{2\times 2} - \mathrm{i}\sum_{c=1}^3\varepsilon_{abc} \sigma_{4-c}, $$
$$ \qquad\qquad a,b~\in~\{1,2,3\},\tag{2}$$
with complex unit $\mathrm{i}\in\mathbb{C}$ that commutes with the quaternionic units $i_a$, $a\in~\{1,2,3\}$, cf. e.g this Phys.SE post.
Defining the associative algebra of biquaternions as
$$\mathbb{B}~:=~\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}~\cong~\mathbb{H}\otimes_{\mathbb{R}}\mathbb{C},\tag{3}$$
we evidently have a $\mathbb{C}$-algebra isomorphism
$$\Phi:~~\mathbb{B}~~\longrightarrow ~~{\rm Mat}_{2\times 2}(\mathbb{C}).\tag{4}$$
by extending the definition
$$\begin{align}\Phi(1)~=~&\sigma_{0}~:=~{\bf 1}_{2\times 2},\cr \Phi(i_1)~=~&\mathrm{i}\sigma_3 , \qquad
\Phi(i_2)~=~\mathrm{i}\sigma_2 , \qquad
\Phi(i_3)~=~\mathrm{i}\sigma_1, \cr
\Phi(i_a)~=~&\mathrm{i}\sigma_{4-a}, \qquad a~\in~\{1,2,3\},\end{align}\tag{5}$$
via $\mathbb{C}$-linearity. The biquaternions $\mathbb{B}$ are also isomorphic to the Clifford algebra ${\rm Cl}(1,2;\mathbb{R})$.
For a biquaternion number
$$b~=~b^0+\sum_{a=1}^3b^ai_a~\in~\mathbb{B}, \qquad b^0,b^1,b^2,b^3~\in~\mathbb{C},\tag{6}$$
define the biconjugate$^1$
$$ \bar{b}~:=~b^0-\sum_{a=1}^3b^ai_a \tag{7}$$
and the complex conjugate$^1$
$$ b^{\ast}~:=~b^{0\ast}+\sum_{a=1}^3b^{a\ast}i_a.\tag{8}$$
Note that Hermitian conjugate matrix is
$$ \Phi(b)^{\dagger}~=~\Phi(\bar{b}^{\ast}),\tag{9} $$
while the transposed matrix is
$$ \Phi(b)^t~=~\Phi(b|_{b^2\to-b^2})~=~\Phi(-j\bar{b}j), \tag{10}$$
so that the complex conjugate matrix is
$$ \Phi(b)^{\ast}~=~\Phi(-jb^{\ast}j). \tag{11}$$
The determinant
$$\det\Phi(b)~=~\sum_{\mu=0}^3 (b^{\mu})^2~=~\bar{b}b~=~\beta(b,b) \tag{12}$$
is the standard bilinear form $\beta:\mathbb{B}\times\mathbb{B}\to \mathbb{C}$
given by
$$ \beta(b,b^{\prime})~:=~\frac{1}{2}(\bar{b}b^{\prime}+\bar{b}^{\prime}b). \tag{13}$$
In other words, the $\mathbb{C}$-algebra isomorphism
$$ (\mathbb{B},\beta)\quad\stackrel{\Phi}{\cong}\quad ({\rm Mat}_{2\times 2}(\mathbb{C}),\det) \tag{14}$$
is an isometry. In fact, the biquaternions $(\mathbb{B},\beta)$ are a composition algebra
$$ \beta(bb^{\prime},bb^{\prime})~=~\det\Phi(bb^{\prime})
~=~\det\Phi(b)\det\Phi(b^{\prime})~=~\beta(b,b) \beta(b^{\prime},b^{\prime}).\tag{15}$$
Moreover
$$\forall b\in\mathbb{B}:~~\beta(b,b)~\neq~0\quad\Leftrightarrow\quad b\text{ is invertible}. \tag{16}$$
The inverse element is
$$b^{-1}~=~\frac{\bar{b}}{\beta(b,b)}.\tag{17}$$
We identify Minkowski space
$$\mathbb{B}~\supseteq\quad\mathbb{R}^{1,3}~:=~\{b\in\mathbb{B}\mid b=\bar{b}^{\ast}\}\quad\stackrel{\Phi}{\cong}\quad u(2)\quad\subseteq~{\rm Mat}_{2\times 2}(\mathbb{C})\tag{18}$$
with the space of $2\times2$ Hermitian matrices. Notice that
the bilinear form $\beta$ restricted to Minkowski space is the standard Minkowski metric with signature (1,3). The biquaternions $\mathbb{B}$ are then the complexified Minkowski space.
We also identify the Lie group
$$\mathbb{B}~\supseteq\quad O(1,\mathbb{B})~:=~\{g\in\mathbb{B}\mid \bar{g}g=1\}\quad\stackrel{\Phi}{\cong}\quad SL(2,\mathbb{C})\quad\subseteq~{\rm Mat}_{2\times 2}(\mathbb{C})\tag{19}$$
with $SL(2,\mathbb{C})$, which is (the double cover of) the restricted Lorentz group $SO^+(1,3;\mathbb{R})$, cf. e.g. this Phys.SE post.
The latter is because there is a group action $\rho:O(1,\mathbb{B})\times \mathbb{R}^{1,3}\to \mathbb{R}^{1,3}$ given by
$$g\quad \mapsto\quad\rho(g)b~:= ~gb\bar{g}^{\ast},
\qquad g\in O(1,\mathbb{B}),\qquad b\in \mathbb{R}^{1,3}, \tag{20}$$
which is length preserving,
$$\begin{align} \overline{\rho(g)b}\rho(g)b
~=~&\overline{gb\bar{g}^{\ast}}gb\bar{g}^{\ast}
~=~g^{\ast}\bar{b}\bar{g}gb\bar{g}^{\ast}
~=~g^{\ast}\underbrace{\bar{b}b}_{\in\mathbb{C}}\bar{g}^{\ast}\cr
~=~&\bar{b}bg^{\ast}\bar{g}^{\ast}
~=~\bar{b}b(g\bar{g})^{\ast}
~=~\bar{b}b, \end{align}\tag{21}$$
i.e. $g$ is a pseudo-orthogonal (or Lorentz) transformation.
The corresponding Lie algebra is
$$\mathbb{B}~\supseteq\quad o(1,\mathbb{B})
~:=~\{b\in\mathbb{B}\mid \bar{b}+b=0\}
\quad\stackrel{\Phi}{\cong}\quad
sl(2,\mathbb{C})\quad\subseteq~
{\rm Mat}_{2\times 2}(\mathbb{C}).\tag{22}$$
To define the left and right Weyl spinor representations we need a fiducial Lie algebra element $b\in o(1,\mathbb{B})$ that is nilpotent $b^2=0$, and a corresponding projection $p\propto \bar{b}^{\ast}b$, i.e. $$\bar{p}^{\ast}~=~p~=~p^2,\tag{23}$$
cf. Ref. 1. To be concrete we choose the nilpotent elements
$$\begin{align} \sigma_{\mp}^{\dagger}~=~\sigma_{\pm}~:=~&\frac{\sigma_1\pm\mathrm{i}\sigma_2}{2}~=~\Phi(b_{\pm})~\in~sl(2,\mathbb{C}), \cr
b_{\pm}~:=~&\frac{-\mathrm{i}k\pm j}{2}~\in~o(1,\mathbb{B}), \tag{24}\end{align}$$
and
$$\begin{align} {\bf 1}_{2\times 2}-P_{\mp}~=~P_{\pm}~:=~&\frac{{\bf 1}_{2\times 2}\pm\sigma_3}{2}~=~\Phi(p_{\pm})~\in~u(2),\cr
p_{\pm}~:=~&\frac{1\mp\mathrm{i}i}{2}~\in~\mathbb{R}^{1,3}.
\tag{25}\end{align}$$
We next define left and right Weyl spinor representations
$$\begin{align}\rho_{\pm}: O(1,\mathbb{B})\times V_{\pm}~\to~&V_{\pm},\cr \rho_+(g)\psi_+~:=~&g\psi_+,\qquad
\rho_-(g)\psi_-~:=~-jg^{\ast}j\psi_-,\cr
g~\in~&O(1,\mathbb{B}), \qquad
\psi_{\pm}~\in~ V_{\pm}, \tag{26}\end{align}$$
where the $\mathbb{C}$-vector spaces are given by
$$ V_{\mp}^{\ast}~=~V_{\pm}
~:=~\mathbb{B}p_{\pm}
~=~\mathbb{B}b_{\mp}
~=~{\rm span}_{\mathbb{C}}\{p_{\pm},b_{\mp}\}
~\subseteq~\mathbb{B}.
\tag{27} $$
The Dirac spinor space is the biquaternions
$$ V_+\oplus V_-~=~\mathbb{B}, \tag{28} $$
while the Majorana spinor space is the quaternions
$$\{q\in\mathbb{B}\mid q=q^{\ast}\}~\cong~\mathbb{H}.\tag{29}$$
