Discrete logarithm index calculus algorithms are usually more efficient for non-hyperelliptic curves (Diem's method) than for hyperelliptic curves (Gaudry's method). However when the field of definition is not prime, Nagao's algorithm is even faster asymptotically, but is more efficient for hyperelliptic curves than for non-hyperelliptic ones. A natural question is then whether it is possible to adapt Nagao's method and design an index calculus that takes advantage of both the field extension and the non-hyperellipticity. In this work we explain why this is not possible, and why the asymptotic complexity of Nagao's algorithm is optimal using the known decomposition techniques.