There were experiments in the late 90's to "thread" the 3rd dimension back along the paths of motion vectors, but this gets complicated in a hurry (consider two different motion vectors that point back to the same pixel), and never really demonstrated compelling advantages.
Complexity is the one thing that's actually not a problem. Most practical transforms are separable, meaning you can, e.g., implement the 2D transform by applying a 1D transform to the rows and then a 1D transform to the columns. So the cost is linear in the number of dimensions.
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