  We study the dynamics of two level systems described by non-hermitian
Hamiltonians with real eigenvalues. Within the framework of hermitian quantum
mechanics, it is known that maximal violation of Leggett-Garg inequality is
bounded by $3/2$ (Luder's bound). We show that this absolute bound can be
evaded when dynamics is governed by non-hermitian Hamiltonians. Moreover, the
extent of violation can be optimized to reach its algebraic maximum of $3$
which is otherwise only feasible when the Hilbert space is infinite dimensional
in the hermitian case. The extreme violation of Leggett-Garg inequality is
shown to be directly related to the two basic ingredients: (i) The Bloch
equation for the two level system has non-linear terms which allow for
accelerated dynamics of states on the Bloch sphere exceeding all known quantum
speed limits of state evolution; and (ii) We need to ensure that the quantum
trajectory of states always lies on a single great circle (geodesic path) on
the Bloch sphere at all times.
