  Recent results of Freitas, Kraus, Sengun and Siksek give sufficient criteria
for the asymptotic Fermat's Last Theorem to hold over various number fields. In
this paper, we prove asymptotic results about the solutions of the Diophantine
equation $x^4-y^4=nz^p$ over various number fields using the modular method.
For instance, we prove that the asymptotic generalised Fermat Theorem for the
equation $x^4-y^4=2^\alpha z^p$ holds for infinitely many quadratic number
fields.
