Some Test Cases

OscillatorPlot[Exp[-x^2-I x],{x},PlotStyle->Thickness[0.05],PlotLabel->"Erster Versuch", AccuracyGoal->4, PrecisionGoal->4]//Timing

OscillatorPlot[(E^(I x - 2  (x^2))), {x, -5, 5},  {0, 8}, PlotStyle -> Thickness[0.08],  
      Axes -> True,  Frame -> False]