To handle quadratic edge cases you would switch to heapsort when the recursion depth goes above the threshold. I.e. introsort; no need for convoluted pivot selection methods.
That's usually called Introsort, but I think you'll find that most of these still consider at least the middle element for a pivot, so they still have a more complex cache behaviour than the naïve linear sweep from both ends of a simple Quicksort.
