  Let $t$ be any imaginary quadratic integer with $|t|\geq 100$. We prove that
the inequality \[
  |F_t(X,Y)|
  = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 |
  \leq 1 \] has only trivial solutions $(x,y)$ in integers of the same
imaginary quadratic number field as $t$. Moreover, we prove results on the
inequalities $|F_t(X,Y)| \leq C|t|$ and $|F_t(X,Y)| \leq |t|^{2 -\varepsilon}$.
These results follow from an approximation result that is based on the
hypergeometric method. The proofs in this paper require a fair amount of
computations, for which the code (in Sage) is provided.
