 Solution

The components are in phase quadrature when the axes of the ellipse coincide with the coordinate axes. Since the $x$ wave amplitude is one and the $y$ wave amplitude is two, the Jones vector must have $A=\pm 2i$.

A handedness convention is essential. Adopt the optical convention in which right-handed elliptical polarization rotates clockwise when viewed toward the source, from the positive-$z$ side. With the paper's $e^{+i\omega t}$ phase convention for complex amplitudes, this givesThe tip follows ${\Psi }_x^2+{\Psi }_y^2/4=1$. In particular,

The figure states the viewing direction and shows increasing time. Under the alternative convention that calls rotation from $+x$ toward $+y$ right-handed when looking along propagation, the equally consistent answer is $A=-2i$ and the second vector is $(0,+2)$. The original paper does not state a handedness convention, so its wording alone cannot select between these labels; the real components and viewing direction remove the ambiguity.