  The present paper introduces a quantum weak measurement (WM) scheme for the
measurement of the complex magneto-optical Kerr effect (MOKE). We achieve the
simultaneous measurement of the Kerr rotation angle and the ellipticity in a
single WM process by utilizing two auxiliary pointers derived from the same
meter state. The experimental measurement precision for both the Kerr rotation
angle and the ellipticity is capable of reaching $10^{-4}$ deg. This technique
is also employed for the determination of the complex magneto-optical constant
$Q$. The proposed method overcomes the limitation of acquiring the complex
magneto-optical Kerr parameters through a multi-step measurement process, which
was previously encountered. This breakthrough holds immense significance for
efficiently measuring and applying the complex MOKE with high precision and
cost-effectiveness.
