  In cryptography, we hope a sequence over $\mathbb{Z}_m$ with period $N$
having larger $m$-adic complexity. Compared with the binary case, the
computation of 4-adic complexity of knowing quaternary sequences has not been
well developed. In this paper, we determine the 4-adic complexity of the
quaternary cyclotomic sequences with period 2$p$ defined in [6]. The main
method we utilized is a quadratic Gauss sum $G_{p}$ valued in
$\mathbb{Z}_{4^N-1}$ which can be seen as a version of classical quadratic
Gauss sum. Our results show that the 4-adic complexity of this class of
quaternary cyclotomic sequences reaches the maximum if $5\nmid p-2$ and close
to the maximum otherwise.
