Regular complete permutation polynomials over quadratic extension fields

Let $r\ge 3$ be any positive integer which is relatively prime to $p$ and $q^2\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}r)$. 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.