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 $r otin lbrace p,qrbrace$. 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 $bigl(tfrac{r}{p}bigr) = bigl(tfrac{r}{q}bigr) = -1$, $r mid 3pq-1$, and $r mid 2pq-4$, then the linear complexity can reach the maximal value equal to the length of the sequences.
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