  In this article we study the strong unique continuation property for
solutions of higher order (variable coefficient) fractional Schr\"odinger
operators. We deduce the strong unique continuation property in the presence of
subcritical and critical Hardy type potentials. In the same setting, we address
the unique continuation property from measurable sets of positive Lebesgue
measure. As applications we prove the antilocality of the higher order
fractional Laplacian and Runge type approximation theorems which have recently
been exploited in the context of nonlocal Calder\'on type problems. As our main
tools, we rely on the characterisation of the higher order fractional Laplacian
through a generalised Caffarelli-Silvestre type extension problem and on
adapted, iterated Carleman estimates.
