Block[{x0, y0, eqns, soln},
 x0 = -1;
 y0 = 1;
 eqns = Join[
   Thread[{x'[s], y'[s]} == ({Re[#], Im[#]}/Abs[#])&[((x[s] - I y[s])^2)^(1/3)]],
   Thread[{x[0], y[0]} == {x0, y0}]
   ];
 soln = First@NDSolve[eqns, {x, y}, {s, 0, 3}];

 ParametricPlot[
  {x[s], y[s]} /. soln
  , {s, 0, 3}
  , Frame -> True
  ]
 ]