  Suppose that $f$ is a $K$-quasiconformal self-mapping of the unit disk
$\mathbb{D}$, which satisfies the following: $(1)$ the biharmonic equation
$\Delta(\Delta f)=g$ $(g\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) the
boundary condition $\Delta f=\varphi$ ($\varphi\in\mathcal{C}(\mathbb{T})$ and
$\mathbb{T}$ denotes the unit circle), and $(3)$ $f(0)=0$. The purpose of this
paper is to prove that $f$ is Lipschitz continuos, and, further, it is
bi-Lipschitz continuous when $\|g\|_{\infty}$ and $\|\varphi\|_{\infty}$ are
small enough. Moreover, the estimates are asymptotically sharp as $K\to 1$,
$\|g\|_{\infty}\to 0$ and $\|\varphi\|_{\infty}\to 0$, and thus, such a mapping
$f$ behaves almost like a rotation for sufficiently small $K$, $\|g\|_{\infty}$
and $\|\varphi\|_{\infty}$.
