I find a variety of definitions for “Unimodular” on the web as of 2007 Mar 15. They require a square matrix over:
Planet Math (P)
a some field and determinant = 1.
Wikipedia (K)
Z = the integers, with determinant = ±1.
Wolfram (W)
reals and det = 1 but generalizes by allowing elements from some polynomial domain if the matrix has an inverse in that domain.
(This suggests ring elements, requiring inverse matrix over same ring.)

Analysis, Manifolds and Physics (page 173 in my edition) specifies real entries and det=1.

Baez means complex matrices A where A*A=I and det(A) = 1.

is unimodular for K and W but not P.
is unimodular for P and W but not K.
is unimodular for P but neither W or K.

All of these are groups but are generalizations in roughly two different directions. In physics contexts it seems to be a real matrix with det=1.

The major split I see is between the discrete and continuous.

When the matrices are over a field and det=1 then this code computes a matrix inverse. Regarding the permitted values of the determinant, any multiplicative subgroup of a field will do. The resulting set of matrices will be a group. The unit circle in the complex numbers is an interesting case. In rings ‘unit’ refers to any factor of 1, such as −1 in (−1)(−1) = 1. i is a unit within the integral complex numbers (Gaussian integers).
If we start with matrices over a ring and demand that the determinant belong to some particular multiplicative subgroup of the ring, then we still can’t count on finding an inverse, but we can require it as a condition of membership in our group of matrices.
Here is a generalization that covers all of these cases:
For any ring the set of n by n matrices over the ring with an inverse over the ring form a group M. If U is a multiplicative subgroup of the units of the ring then those matrices with determinant in U form a subgroup of M. (For example {−1, 1} and {1} are the two subgroups of the units of the ring of integers.) N.B. if the ring has no unit then it has no multiplicative groups. Adjusting the ring and the group U gives us all the concepts of ‘unimodular’ that I have found online.