Drag z around the left plane — especially in a loop around the origin — and watch its several values f(z) move in the right plane. The principal branch is red, the others blue. Circle the origin once and the branches swap places: that's the branch cut in action.
A modern JavaScript + Canvas port of a 1998 Java applet by Terence Tao
(originally MultiApp.java / MultiGrid.java).
Drag z in a loop around 0 to see the sheets permute.