In upper half space, the thing that's analogous to the invariant 2-form (which measures hyperbolic area) is the hyperbolic volume form $1/z^3 dx dy dz$.

If you want to understand 2-forms invariant by a particular discrete group, they are
the same thing as 2-forms on the quotient orbifold. In the case of $SL_2$ of the algebraic integers in a quadratic imaginary field such as this, the groups are usually called Bianchi
groups: Bianchi worked out fundamental domains for them for the first set of examples, as well as a general technique to find fundamental domains.  On Allen Hatcher's website, there's a [handy list][1] of pictures of the folded-up fundamental domains in cases where they can be
readily drawn. 

In this case, the quotient orbifold has underlying space $S^3$ minus a point
(for the cusp), with singular locus in the form of the 1-skeleton of a tetrahedron. Since the underlying space is contractible, any closed 2-form is a coboundary --- *i.e.*
there are plenty of 2-forms, closed or otherwise, but there's no obvious reason they
should be interesting. In this particular
case, the diagonal matrix with entries $i, -i$ rotates 180 degrees about the
$z$-axis, acting as an orientation-reversing map on the hyperbolic plane, so it takes the 2-form above to its negative: it's not invariant even
before attempting to extend it.


  [1]: http://www.math.cornell.edu/~hatcher/Papers/Bianchi.pdf