Accurate asymptotic formulas are obtained for the eigenvalues and eigenfunctions of the nonself-adjoint fourth-order periodic boundary value problem \[ \begin{aligned} &u^{(4)}(t)+q(x)y=\lambda y, \quad 0\leq x\leq \pi,\\ &y^{(j)}(0)-y^{(j)}(\pi)=0,\quad j=0, 1, 2, 3, \end{aligned} \] where \(q(x)\) is a complex valued function satisfying \(\int_0^{\pi}q(x)\, dx=0.\)