  In this short paper, we prove that the solution of the cubic fourth-order
Schr\"odinger equation (4NLS) on $\mathbb{R}^d$ ($5 \leq d \leq 8$) enjoys the
same (pointwise) decay property as its linear solution does. This result is
proved via a bootstrap argument based on the corresponding global result
Pausader \cite{Pau1}. This result can be extended to more general dispersive
equations (including some more 4NLS models) with scattering asymptotics.
