The reason we want to switch from the coefficient coordinate system to Hermite basis system is to switch to the coordinate system of [p0, p1, u0, u1].

I wonder what is the advantage of using Hermite Basis over the common basis?

In this basis, the points p_0 and p_1 can be 2D? But the derivatives would be 1D?

Or I guess what you could do is if you have points with different x-coordinates, t would basically interpolate between those for each pair of points.