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Arithmetic-geometric mean sequences were already studied over the real and complex numbers, and recently, Griffin et al. [‘AGM and jellyfish swarms of elliptic curves’, Amer. Math. Monthly130(4) (2023), 355–369] considered them over finite fields $\mathbb {F}_q$ for $q \equiv 3 \pmod 4$. We extend the definition of arithmetic-geometric mean sequences over $\mathbb {F}_q$ to $q \equiv 5 \pmod 8$. We explain the connection of these sequences with graphs and explore the properties of the graphs in the case where $q \equiv 5 \pmod 8$.
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