<<Cauchy.m
f = z^8 - 5 z^7 + 7 z^6 - 12 z^5 + 14 z^4 - 12 z^3 + 7 z^2 - 5 z + 1
PlotRootsContour[f, z, 
                 ComplexTriangle[-I, 4+3 I, t], {t, 0, 3}]
G = PlotArgumentPrinciple[f, z,
                          ComplexTriangle[-I, 4+3 I, t], {t, 0, 3},
                          ViewPoint->{0,0,100}]
(* shown contour around -I blown-up *)
G2 = PlotArgumentPrinciple[f, z,
                          ComplexTriangle[-I, 4+3 I, t], {t, 0, 1/5},
                          DisplayFunction->Identity]
G3 = PlotArgumentPrinciple[f, z,
                          ComplexTriangle[-I, 4+3 I, t], {t, 3-1/5, 3},
                          DisplayFunction->Identity]
Show[{G2, G3}, ViewPoint->{0,0,100}, DisplayFunction->$DisplayFunction]