The components of a two (complex) dimensional Dirac system are studied as trajectories in the complex plane. The system, defined on an interval [a, b) of regular points, is assumed to be of limit circle type at t = b, and to be non-oscillatory there. By introducing moduli ρ1(t) = |y1(t)|, ρ2(t) = |y2(t)| and continuous complex arguments θ1(t) = arg y1(t), θ2(t) = arg y2(t) for the components, the principal result proved is that θ1(t) and θ2(t) are bounded as t → b. Examples show that monotonicity of the argument function θ1(t), which is a feature of the corresponding problem for Sturm–Liouville equations, fails for Dirac systems.