Quartic Equations with Trivial Solutions over Gaussian Integers

In our work we study the equations of the form $a{X}^4+b{X}^2{Y}^2+c{Y}^4=d{Z}^2$ 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 $\mathbb{Z}[i]$), $X^4+6(1+i)X^2{Y}^2+2i{Y}^4=Z^2$ 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), $X^4\pm Y^4=i{Z}^2$ (Szab'o - Najman).