By the OP's request, I collect my comments into an answer.

1. The usual left-action of $\mathrm{SL}_2(\mathbb{C})$ on $\text{H}^{+}_3$ is defined by quaternionic right-division: $MP:=(\alpha P+\beta)(\gamma P+\delta)^{-1}$. Here $P=z+wj$ is a Hamiltonian quaternion with $k$-component zero, and so is $MP$. This left-action induces a right-action on functions $\text{H}^{+}_3\to\mathbb{C}$: the image of the function $P\mapsto f(P)$ is $P\mapsto f(M P)$. It also induces a left-action on functions $\text{H}^{+}_3\to\mathbb{C}$: the image of the function $P\mapsto f(P)$ is $P\mapsto f(M^{-1} P)$. These group actions (on functions $\text{H}^{+}_3\to\mathbb{C}$) give rise to Lie-algebra actions (on smooth functions $\text{H}^{+}_3\to\mathbb{C}$) in the usual way. Note that smooth and compactly supported functions are dense in $\mathcal{L}^2(\text{H}^{+}_3)$.

2. Both the left and the right action are unitary, because $\mathrm{SL}_2(\mathbb{C})$ is a unimodular group, hence it has a two-sided Haar measure. The Lie-algebra action is as follows. An element $m\in\mathfrak{sl}_2(\mathbb{C})$ maps the function $P\mapsto f(P)$ to the function $P\mapsto\frac{\partial}{\partial t}f(e^{-t m}P)\mid_{t=0}$. Of course this action is only defined for special functions $P\mapsto f(P)$, e.g. smooth and compactly supported functions are fine.

3. The space $\text{H}^{+}_3$ can be naturally identified with $\mathrm{SL}_2(\mathbb{C})/\mathrm{SU}_2(\mathbb{C})$; this identifaction comes from the left action of $\mathrm{SL}_2(\mathbb{C})$ on $\text{H}^{+}_3$. Hence a natural measure on $\text{H}^{+}_3$ is induced by a right Haar measure on $\mathrm{SL}_2(\mathbb{C})$. However, this right Haar measure is also a left Haar measure, whence the natural measure on $\text{H}^{+}_3$ is in fact invariant under the left action of $\mathrm{SL}_2(\mathbb{C})$. So this left action induces a unitary action on $\mathcal{L}^2(\text{H}^{+}_3)$.

By the OP's request, I collect my comments into an answer.

1. The usual left-action of $\mathrm{SL}_2(\mathbb{C})$ on $\text{H}^{+}_3$ is defined by quaternionic right-division: $MP:=(\alpha P+\beta)(\gamma P+\delta)^{-1}$. Here $P=z+wj$ is a Hamiltonian quaternion with $k$-component zero, and so is $MP$. This left-action induces a right-action on functions $\text{H}^{+}_3\to\mathbb{C}$: the image of the function $P\mapsto f(P)$ is $P\mapsto f(M P)$. It also induces a left-action on functions $\text{H}^{+}_3\to\mathbb{C}$: the image of the function $P\mapsto f(P)$ is $P\mapsto f(M^{-1} P)$. These group actions (on functions $\text{H}^{+}_3\to\mathbb{C}$) give rise to Lie-algebra actions (on smooth functions $\text{H}^{+}_3\to\mathbb{C}$) in the usual way. Note that smooth and compactly supported functions are dense in $\mathcal{L}^2(\text{H}^{+}_3)$.

2. Both the left and the right action are unitary, because $\mathrm{SL}_2(\mathbb{C})$ is a unimodular group, hence it has a two-sided Haar measure. The Lie-algebra action is as follows. An element $m\in\mathfrak{sl}_2(\mathbb{C})$ maps the function $P\mapsto f(P)$ to the function $P\mapsto\frac{\partial}{\partial t}f(e^{-t m}P)\mid_{t=0}$. Of course this action is only defined for special functions $P\mapsto f(P)$, e.g. smooth and compactly supported functions are fine.

3. The space $\text{H}^{+}_3$ can be naturally identified with $\mathrm{SL}_2(\mathbb{C})/\mathrm{SU}_2(\mathbb{C})$; this identification comes from the left action of $\mathrm{SL}_2(\mathbb{C})$ on $\text{H}^{+}_3$. Hence a natural measure on $\text{H}^{+}_3$ is induced by a right Haar measure on $\mathrm{SL}_2(\mathbb{C})$. However, this right Haar measure is also a left Haar measure, whence the natural measure on $\text{H}^{+}_3$ is in fact invariant under the left action of $\mathrm{SL}_2(\mathbb{C})$. So this left action induces a unitary action on $\mathcal{L}^2(\text{H}^{+}_3)$.