  Let $r\geq 3$ be any positive integer which is relatively prime to $p$ and
$q^2\equiv 1 \pmod r$. Let $\tau_1, \tau_2$ be any permutation polynomials over
$\mathbb{F}_{q^2},$ $\sigma_M$ is an invertible linear map over
$\mathbb{F}_{q^2}$ and $\sigma=\tau_1\circ\sigma_M\circ\tau_2$. In this paper,
we prove that, for suitable $\tau_1, \tau_2$ and $\sigma_M$, the map $\sigma$
could be $r$-regular complete permutation polynomials over quadratic extension
fields.
