Does the determinant of a complex-valued matrix have a geometric interpretation? The determinant of a real-valued matrix can be seen as the volume of the parallelotope with the column vectors as the sides.  Is there an analogous interpretation for complex-valued matrix determinants?  I'm not entirely sure what a complex-valued vector would look like and thus how to build a parallelotope out of one, but maybe there's some other interpretation.
Looking at some similar questions, I'm wondering if Clifford algebra could be the key here?  I'm not sure.  I don't think I've worked with complex spaces enough to be able to figure it out on my own.
Any thoughts?
 A: Ok if nobody else is going to address this one let me give it a try. 
In complex numbers the real and imaginary components operate by different
rules.  Multiplication by the real component scales a vector in or out from the
origin, whereas multiplication by the imaginary component rotates it about the
origin.
In Clifford Algebra there is no distinction between real and imaginary
components, all dimensions work by the same rules. Multiplying parallel vectors
scales their length in or out from the origin, whereas multiplication of
non-parallel vectors results in a bivector product with a "twist" that rotates
through the angle from the first vector to the second.
The volume of the parallelotope due to multiplication of the row-vectors uses
"real" multiplication, i.e. without an imaginary component. The Wiki page on
the determinant says 
"The bivector magnitude (denoted (a, b) ∧ (c, d)) is the signed area, which is
also the determinant ad − bc."
http://en.wikipedia.org/wiki/Determinant
I presume the answer to your question about a "complex" determinant would be
the bivector itself, not just the "bivector magnitude", i.e. not just the area
of the parallelogram in the 2-D example, but also its "twist", the rotational
component of the multiplication. 
As to what that actually "means" is open to interpretation. The utility of the
determinant is what it tells you about the matrix, or how it "represents the
image of the unit square under the mapping" (from the Wiki page). So I presume
the "complex determinant" tells you not only how the matrix would scale the
unit square, but how it would twist it aswell.
