Calculating the resulting phase from its complex $x$ and $y$ field components

I am going to assume that your three light sources emit coherent waves which combine to produce oscillating E field components at each point in the region, and the phase of these oscillations (relative to some chosen reference time) is represented by vectors rotating in the complex number plane. The A,B,C, and D represent instantaneous values for the components of these vectors. These imaginary vectors are similar to those used to represent voltages in an AC circuit. The magnitude of the oscillations of the electric field components at each point would be given by the square root of the real part of the phase vector squared plus the imaginary part squared, and the phase by the arc-tangent as you suggested. The resultant electric field vector would rotate the xy plane. If the phase difference between the two electric field components is $90^o$, the resultant vector will sweep out an ellipse. If the two are in phase, the result is a straight line. Other phase differences give intermediate shapes.