Let $r\geq 3$ be any positive integer which is relatively prime to $p$ and $q^2\equiv 1 \pmod r$. Let $τ_1, τ_2$ be any permutation polynomials over $\mathbb{F}_{q^2},$ $σ_M$ is an invertible linear map over $\mathbb{F}_{q^2}$ and $σ=τ_1\circσ_M\circτ_2$. In this paper, we prove that, for suitable $τ_1, τ_2$ and $σ_M$, the map $σ$ could be $r$-regular complete permutation polynomials over quadratic extension fields.