  We construct the $N=2$ super $W_4$ algebra as a certain reduction of the
second Gel'fand-Dikii bracket on the dual of the Lie superalgebra of $N=1$
super pseudo-differential operators. The algebra is put in manifestly $N=2$
supersymmetric form in terms of three $N=2$ superfields $\Phi_i(X)$, with
$\Phi_1$ being the $N=2$ energy momentum tensor and $\Phi_2$ and $\Phi_3$ being
conformal spin $2$ and $3$ superfields respectively. A search for integrable
hierarchies of the generalized KdV variety with this algebra as Hamiltonian
structure gives three solutions, exactly the same number as for the $W_2$
(super KdV) and $W_3$ (super Boussinesq) cases.
