Needs["NDSolve`FEM`"]
NDSolve[{2 f[x, y] + I*D[f[x, y], y, y] - I*D[f[x, y], x, x] -
2 x*D[f[x, y], y] + 2 x*D[f[x, y], x] + 2 y*D[f[x, y], y] -
2 y*D[f[x, y], x] + 2*f[x, y]*x^2 - 4 f[x, y]*x*y +
2*f[x, y]*y^2 - 2 I*x*D[f[x, y], y] - 2 I*x*D[f[x, y], x] +
2 I*y*D[f[x, y], y] + 2 I*y*D[f[x, y], x] == 0,
f[x, 0] == Exp[-x], f[x, 1] == 1}, f, {x, 0, 1}, {y, 0, 1}]

NDSolve::femdpop: The FEMStiffnessElements operator failed.