  Let ${\mathcal M}_2(\mathbb F)$ be the algebra of 2$\times$2 matrices over
the real or complex field $\mathbb F$. For a given positive integer $k\geq 1$,
the $k$-commutator of $A$ and $B$ is defined by $[A,B]_k=[[A,B]_{k-1},B]$ with
$[A,B]_0=A$ and $[A,B]_1=[A,B]=AB-BA$. The main result is shown that a map
$\Phi: {\mathcal M}_2(\mathbb F)\to {\mathcal M}_2(\mathbb F)$ with range
containing all rank one matrices satisfies that $[\Phi(A),\Phi(B)]_k = [A,B]_k
$ for all $A, B\in{\mathcal M}_2(\mathbb F)$ if and only if there exist a
functional $h :{\mathcal M}_2(\mathbb F) \rightarrow {\mathbb F}$ and a scalar
$\lambda \in{\mathbb F}$ with $\lambda^{k+1} = 1$ such that $\Phi(A) = \lambda
A + h(A)I$ for all $A \in{\mathcal M}_2(\mathbb F)$.
