  This paper discusses the unique continuation principal of the solutions of
the following perturbed fourth order elliptic differential operator
$\mathcal{L}_{A,q}u=0$, where \[ \mathcal{L}_{A,q}(x,D)\ =\
\sum_{j=1}^nD^4_{x_j} + \sum_{j=1}^n A_jD_{x_j} + q, \qquad (A, q) \in
W^{1,\infty}(\Omega,\mathbb{C}^n) \times L^{\infty}(\Omega,\mathbb{C}) \] whose
principal term is not given by some integer power of the Laplacian operator. We
derive some suitable Carleman estimates which is the main tool to prove the
unique continuation principle. As a by-product, we also deduce some stability
estimate and prove the strong unique continuation principle in $2$-dimension.
