Ever since I first took Linear Algebra, I have over time realized how concepts like determinants, eigenvalues, diagonalization, orthogonal transformations and so on have very intuitive geometric interpretations, and so begun to fill in many gaps in my understanding of the subject (the course I took, just like most freshman-year courses I've taken, focused too much on pure computation and not enough on intuition / visualization, as do most course books on linear algebra that I've read).
Now, determinants are volumes (technically, multiplicative volume changes), diagonalization is just re-expressing the action of a matrix in terms of scalings of an eigenbase, and orthogonal transformations are norm- (and inner product-) preserving, and therefore fully characterized by compositions of rotations and reflections (and I still don't fully understand why the books start with $A^{-1}=A^T$ as the first definition of an orthogonal matrix…)…
…but what about Unitary transformations / matrices? They are also norm- (and inner product-) preserving, but may take complex-valued entries. How can I interpret this geometrically? For example, I know that the group SU(2) can be visualized as the 3-sphere (they are apparently diffeomorphic), but I cannot see the picture clearly in my head.