We consider the anti-de Sitter space $H_3^1$ and the hyperbolic Hopf fibration $h : H_3^1 (1) o H_2(1)$. Using their description in terms of paraquaternions, we study the magnetic curves of the hyperbolic Hopf vector field. A complete classification is obtained for light-like magnetic curves, showing in particular the existence of periodic examples, and emphasizing their relationship with the hyperbolic Hopf fibration. Finally, we give a new interpretation of magnetic curves in $H^3_1$ using some techniques of Lie groups and Lie algebras.
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