  In this paper, the linear complexity over $\mathbf{GF}(r)$ of generalized
cyclotomic quaternary sequences with period $2pq$ is determined, where $ r $ is
an odd prime such that $r \ge 5$ and $r\notin \lbrace p,q\rbrace$. The minimal
value of the linear complexity is equal to $\tfrac{5pq+p+q+1}{4}$ which is
greater than the half of the period $2pq$. According to the Berlekamp-Massey
algorithm, these sequences are viewed as enough good for the use in
cryptography. We show also that if the character of the extension field
$\mathbf{GF}(r^{m})$, $r$, is chosen so that $\bigl(\tfrac{r}{p}\bigr) =
\bigl(\tfrac{r}{q}\bigr) = -1$, $r\nmid 3pq-1$, and $r\nmid 2pq-4$, then the
linear complexity can reach the maximal value equal to the length of the
sequences.
