Context: A common visualization of the meaning of a (real) determinant is the area (or volume, in higher dimensions) enclosed by the image of basis vectors under matrix transformation:
Question: But what if the matrix and determinant in question are complex valued? How do we visualize the meaning of the determinant of these cases?
Example: For example, the determinant of a unitary matrix $U$ satisfies
$$ \det(U) = e^{i\phi} $$
for some $\phi$. Is there a way to visualize what the complex determinant of $U$ is, in this case?