  In our work we study the equations of the form $aX^4+bX^2 Y^2+cY^4=dZ^2$ over
Gaussian integers by a method of the resolvents. We study as a new equations
$X^4+6X^2 Y^2+Y^4=Z^2$ (Mordell's equation over $\mathbb{Z}[i]$),
$X^4+6(1+i)X^2Y^2+2iY^4=Z^2$ and $X^4\pm Y^4=(1+ i)Z^2$ and give the new proofs
of the known theorems on $X^4+Y^4=Z^2$ (Fermat - Hilbert), $X^4\pm Y^4=iZ^2$
(Szab\'o - Najman).
