  We study the fourth order Schr\"odinger operator $H=(-\Delta)^2+V$ for a
decaying potential $V$ in four dimensions. In particular, we show that the
$t^{-1}$ decay rate holds in the $L^1\to L^\infty$ setting if zero energy is
regular. Furthermore, if the threshold energies are regular then a faster decay
rate of $t^{-1}(\log t)^{-2}$ is attained for large $t$, at the cost of
logarithmic spatial weights. Zero is not regular for the free equation, hence
the free evolution does not satisfy this bound due to the presence of a
resonance at the zero energy. We provide a full classification of the different
types of zero energy resonances and study the effect of each type on the time
decay in the dispersive bounds.
