  We explore the role of exceptional points and complex eigenvalues on the
occurrence of the quantum Mpemba effect. To this end, we study a two-level
driven dissipative system subjected to an oscillatory electric field and
dissipative coupling with the environment. We find that both exceptional points
and complex eigenvalues can lead to $multiple$ quantum Mpemba effect. It occurs
in an observable when time evolved copies corresponding to two different
initial conditions, one initially having higher observable value compared to
the other and both relaxing towards the same steady state, intersect each other
more than once during their relaxation process. Each of the intersections
denotes a quantum Mpemba effect and marks the reversal of identities between
the two copies i.e. the copy with higher observable value before the
intersection becomes the lower valued copy (and vice versa) after the
intersection. Such multiple intersections originate from additional algebraic
time dependence at the exceptional points and due to oscillatory relaxation in
the case of complex eigenvalues. We provide analytical results for quantum
Mpemba effect in the density matrix in presence of coherence. Depending on the
control parameters (drive and dissipation), observables such as energy, von
Neumann entropy, temperature etc. exhibit either single or multiple quantum
Mpemba effect. However, the distance from steady state measured in terms of the
Kullback-Leibler divergence shows only single quantum Mpemba effect although
the corresponding speed gives rise to either single or multiple quantum Mpemba
effect.
