x = RandomComplex[{-1 - I, 1 + I}, 1000000];

arg1[x_] := Mod[Arg@x, 2 π];
arg2[x_] := Arg[-x] + π;
arg3[z_] := π + ArcTan[-Re[z], -Im[z]];

Max[Abs[arg1[x] - arg2[x]], Abs[arg1[x] - arg3[x]]]
(* 8.88178*10^-16 *)

arg1[x]; // AbsoluteTiming
(* {0.16715, Null} *)

arg2[x]; // AbsoluteTiming
(* {0.154602, Null} *)

arg3[x]; // AbsoluteTiming
(* {0.090001, Null} *)

Block[{Arg = arg3}, Arg[Exp[-I]]]
(* -1 + 2 π *)

Block[{Arg = arg3}, Plot[Arg@Exp[I φ], {φ, -2 π, 2 π}]]

arg[z_] /; Im[z] < 0 := Arg[z] + 2 Pi; arg[z_] /; Im[z] >= 0 := Arg[z]

argVec[zVec_] := Module[{imShift},
 imShift = Unitize[(Sign /@ Im[zVec] - 1)]*(2π//N);
 Arg[zVec] + imShift];