In this paper, we prove that $1/|\omega|^2$-harmonic quasiconformal mapping is bi-Lipschitz continuous with respect to quasihyperbolic metric on every proper domain of $\mathbb C\backslash\{0\}$. Hence, it is hyperbolic quasi-isometry in every simply connected domain on $\mathbb C\backslash\{0\}$, which generalized the result obtained in [14]. Meanwhile, the equivalent moduli of continuity for $1/|\omega|^2$-harmonic quasiregular mapping are discussed as a by-product.