If I understood my automorphic forms correctly, at least cusp forms can be thought of as elements of $L^2(G/\Gamma)$ for a $G = \text{SL}_2(\mathbb{R})$ and $\Gamma = \text{SL}_2(\mathbb{Z})$ or a suitable congrence subgroup.

I want to know if there is an analogous result for hyperbolic-3 space modulo a Bianchi group. For example: $$ \mathbb{H}^3 \,/\, \text{PSL}_2 \big(\mathbb{Z}[i] \big) $$

This is a group action by isometries. They inherit from this action of the full group:

\begin{eqnarray*} \text{SL}_2(\mathbb{C}) \times \mathbb{H}^3 &=& \mathbb{H}^3\\ (g,z) &=& \frac{az+b}{cz+d} \end{eqnarray*}

In this case, is there a version of modular forms (or cusp forms) here? I have never seen a discussion of modular forms over quaternionic arguments. I suppose the method of images could produce an invariant function:

$$ f(z) = \sum_{\gamma \in \text{PSL}_2(\mathbb{Z}[i])} f_0( \gamma \, z) $$

I am hoping or something more explicit. Is there an analog of theta function or newform in this setting? My guess this should be an element of the space $L^2(G/\Gamma)$ that I have constructed.

$$ f_0(a+bi+cj+dk) = e^{2\pi i \, (a^2 + b^2+c^2 + d^2)} = q^{z \overline{z}} $$

This is just a speculation. Certainly this will be invariant but it won't be in $L^2$.

$\text{PSL}_2(\mathbb{Z}[i])$ is a textbook exmple of a Kleininan group and there are basica ways to construct hyperbolic 3-manifolds, such as $\mathbb{H}^3/\text{PSL}_2(\mathbb{Z}[i])$.

In the case of $\mathbb{H}/\text{SL}_2(\mathbb{Z})$ we could construct modular forms in fairly using Eisenstein series or theta functions. And these examples will be fairly explicit since we often know their Fourier coefficients. And I'm sure the question for modular forms over Bianchi groups is discussed somewhere $\text{PSL}_2(\mathbb{Z}[i])$