  In this paper, we investigate the holographic complexity in the charged
Taub-NUT-AdS black holes with Misner strings present in the Einstein-Maxwell
gravity. We show that differing from the normal black holes, where the
late-time complexity growth rate is only determined by the quantities at outer
and inner ``Reissner-Nordstrom''-type (RN-type) horizons, here the quantities
(the Misner potential and Misner charge) related to the Misner strings also
play an important role in CA complexity. Similar to the case of the normal
electromagnetic black hole, the late-time rate for the original CA conjecture
is independent on the magnetic charges. However, disparate with common results
of the dyonic solutions, the electric charge appeared here is the total charge
of this black hole. Besides, we found that the result in this original CA
conjecture also violates the electromagnetic duality. And this duality can be
restored by adding the Maxwell boundary term with the proportional constant
$\g=1/2$. In this case, the late-time rate is sensitive to the magnetic charge.
Moreover, we also found that the additional term only changes the proportion
between the electric and magnetic charges, and it does not affect the Misner
term appeared in the late-time rate. Finally, we studied the time-dependence of
the complexity growth rate and found that they share similar behaviors with
that in RN-AdS black holes.
