Broadly speaking, linear algebra over the real and the complex fields usually works fairly similarly. Most statements from one have a corresponding statement with the other (with the suitable exchange of the words "orthogonal" and "unitary", "symmetric" and "Hermitian", etc.). One big exception, of course, is eigenvalues: all complex square matrices have an (algebraically) full set of complex eigenvalues, but only exponentially few real square matrices have a full set of real eigenvalues. But there, it's clear that the root of the distinction is that the complex numbers are algebraically closed while the real numbers are not.
There's really only one other qualitative difference that I can think of. For both choices of field, there is are naturally mappings between the set of (bi-/sesqui-)linear forms and the set of quadratic forms on an $n$-dimensional vector space. We can naturally map a (bi-/sesqui-)linear form to a quadratic form by simply plugging the same vector into both slots, and we can go in the other direction using the polarization identity.
But the polarization identity works quite differently for real and complex vector spaces. For a complex vector space, these maps are bijective, and the polarization identity maps a (unique) complex quadratic form to a generic sesquilinear form (not necessarily Hermitian). But for a real vector space, the map from bilinear forms to quadratic forms that comes from restricting to the diagonal is many-to-one: only the symmetric part of the bilinear form gets preserved, while any two bilinear forms that only differ in their antisymmetric part map to the same quadratic form. Moreover, the real version of the polarization identity only maps to the subset of symmetric bilinear forms. So for complex vector spaces, these two maps are inverses, but for real vector spaces they compose to the projection operator down to the proper subspace of symmetric bilinear forms.
What is the fundamental origin of this difference in behavior between the two fields? What structure do the complex numbers have and the reals don't that causes the diagonal of a sesquilinear (but not real bilinear) form to contain enough information to fully reconstruction the form? In this case, it doesn't seem to me like it ultimately stems from the fact that the real numbers aren't algebraically closed, because (unlike with eigenvalues) we only need to do elementary arithmetic operations and never extract any roots.
I think I remember seeing this question on Math SE before, but I can't find it anymore.