Your myLog is discontinuous on the positive real axis:
myLog[z_] := If[Arg[z] < 0, Log[z] + I 2 \[Pi], Log[z]]
FunctionDiscontinuities[myLog[z], z]
ComplexPlot[myLog[z], {z, -1 - I, 1 + I}]
You cannot redefine Log in the complex plane without discontinuities. You may have studied that the vector field (x,-y)/(x^2+y^2) has locally a potential but not globally. It is basically the same thing.
