Probably, you can find a discussion of this in Thurston's notes on hyperbolic 3-manifolds, or maybe some of the expositions by his students.

However, what you are asking for is actually pretty simple: Consider the vector space V of $2$-by-$2$ Hermitian symmetric matrices, thus, $A\in M_2(\mathbb{C})$ lies in $V$ if and only if $A = A^*$, where $A^*$ denotes the conjugate transpose of $A$. The group $\mathrm{SL}(2,\mathbb{C})$ acts on $V$ by $g\cdot A = g A g^*$, and it preserves the (real-valued) quadratic form $Q$ defined by $Q(A) = \det(A)$. Since $$ Q\left(\begin{pmatrix}a+b & z\\ \bar z & a-b\end{pmatrix}\right) = a^2 - b^2 - z\bar z, $$ it follows that the type of $Q$ is $(1,3)$.

Note that this action of $\mathrm{SL}(2,\mathbb{C})$ on $V\simeq\mathbb{R}^{1,3}$ is not effective, since $\pm I_2\in\mathrm{SL}(2,\mathbb{C})$ acts trivially. However, these are the only two elements that act trivially, so this is faithful representation of $\mathrm{PSL}(2,\mathbb{C}) = \mathrm{SL}(2,\mathbb{C})/\{\pm I_2\}$ on $V$. By dimension count, this representation of $\mathrm{PSL}(2,\mathbb{C})$ is the identity component of $\mathrm{O}(Q) \simeq \mathrm{O}(1,3)$.

Now, one can identify hyperbolic $3$-space $H^3$ with the set of matrices in $V$ of determinant $1$ and positive trace, which is the $\mathrm{PSL}(2,\mathbb{C})$-orbit of $I_2\in V$. Alternatively, one can identify hyperbolic $3$-space with the space of (real) $1$-dimensional subspaces of $V$ on which $Q$ is positive definite. In this way, $H^3$ is an open subset of $\mathbb{RP}^3 = \mathbb{P}(V)$, which is an orbit of $\mathrm{PSL}(2,\mathbb{C})$ in its natural action on \mathbb{P}(V)$.

Now consider the set N of $1$-dimensional subspaces of $V$ on which $Q$ vanishes. This is the boundary of hyperbolic space in $\mathbb{RP}^3$. It is not difficult to show that $\ell\in N$ is spanned by a matrix of the form $$ \begin{pmatrix}p\bar p & p\bar q\\ q\bar p & q\bar q\end{pmatrix}. $$ While this does not determine $(p,q)\in\mathbb{C}$ uniquely, it does determine $[p,q]\in\mathbb{CP}^1$, and, vice versa, $[p,q]\in\mathbb{CP}^1$ determines $\ell\in N$.

Thus, $\mathrm{PSL}(2,\mathbb{C})$ acts smoothly and effectively on the space of lines in $\mathbb{RP}^3$ on which $Q$ is non-negative and this is a closed $3$-ball in $\mathbb{RP}^3$ so that the action in the interior is the (identity component of the) isometry group of $H^3$ and the action on the boundary is the standard action of $\mathrm{PSL}(2,\mathbb{C})$ on $\mathbb{CP}^1$.

Probably, you can find a discussion of this in Thurston's notes on hyperbolic 3-manifolds, or maybe some of the expositions by his students.

However, what you are asking for is actually pretty simple: Consider the vector space $V$ of $2$-by-$2$ Hermitian symmetric matrices, thus, $A\in M_2(\mathbb{C})$ lies in $V$ if and only if $A = A^*$, where $A^*$ denotes the conjugate transpose of $A$. The group $\mathrm{SL}(2,\mathbb{C})$ acts on $V$ by $g\cdot A = g A g^*$, and it preserves the (real-valued) quadratic form $Q$ defined by $Q(A) = \det(A)$. Since $$ Q\left(\begin{pmatrix}a+b & z\\ \bar z & a-b\end{pmatrix}\right) = a^2 - b^2 - z\bar z, $$ the type of $Q$ is $(1,3)$.

Note that this action of $\mathrm{SL}(2,\mathbb{C})$ on $V\simeq\mathbb{R}^{1,3}$ is not effective, since $\pm I_2\in\mathrm{SL}(2,\mathbb{C})$ acts trivially. However, these are the only two elements that act trivially, so this is a faithful representation of $\mathrm{PSL}(2,\mathbb{C}) = \mathrm{SL}(2,\mathbb{C})/\{\pm I_2\}$ on $V$. By dimension count, this representation of $\mathrm{PSL}(2,\mathbb{C})$ is the identity component of $\mathrm{O}(Q) \simeq \mathrm{O}(1,3)$.

Now, one can identify hyperbolic $3$-space $H^3$ with the set of matrices in $V$ of determinant $1$ and positive trace, which is the $\mathrm{PSL}(2,\mathbb{C})$-orbit of $I_2\in V$. (Note that the $\mathrm{PSL}(2,\mathbb{C})$-stabilizer of $I_2\in V$ is $\mathrm{SU}(2)/\{\pm I_2\}\simeq \mathrm{SO}(3)$.) Alternatively, one can identify hyperbolic $3$-space with the space of (real) $1$-dimensional subspaces of $V$ on which $Q$ is positive definite. In this way, $H^3$ is an open subset of $\mathbb{RP}^3 = \mathbb{P}(V)$, which is an orbit of $\mathrm{PSL}(2,\mathbb{C})$ in its natural action on $\mathbb{P}(V)$.

Now consider the set $N$ of $1$-dimensional subspaces of $V$ on which $Q$ vanishes. This is the boundary of hyperbolic space in $\mathbb{RP}^3$. It is not difficult to show that $\ell\in N$ is spanned by a matrix of the form $$ \begin{pmatrix}p\bar p & p\bar q\\ q\bar p & q\bar q\end{pmatrix}. $$ While this does not determine $(p,q)\in\mathbb{C}$ uniquely, it does determine $[p,q]\in\mathbb{CP}^1$, and, vice versa, $[p,q]\in\mathbb{CP}^1$ determines $\ell\in N$.

Thus, $\mathrm{PSL}(2,\mathbb{C})$ acts smoothly and effectively on the space of lines in $\mathbb{RP}^3$ on which $Q$ is non-negative, which is a closed $3$-ball in $\mathbb{RP}^3$ so that the action in the interior is the (identity component of the) isometry group of $H^3$ and the action on the boundary is the standard action of $\mathrm{PSL}(2,\mathbb{C})$ on $\mathbb{CP}^1$, i.e., the Riemann sphere.