  We study the dynamics of the Nambu monopole in two Higgs doublet models,
which is a magnetic monopole attached by two topological $Z$ strings ($Z$ flux
tubes) from two opposite sides. The monopole is a topologically stable solution
of the equation of motions when the Higgs potential has global $U(1)$ and
$\mathbb{Z}_2$ symmetries. In this paper, we consider more general cases
without the $\mathbb{Z}_2$ symmetry, and find that it is no longer a static
solution but moves along the $Z$ string being pulled by the heavier string.
After analytically constructing an asymptotic form of the monopole, we confirm
such a motion using the numerical relaxation method. In addition, we analyze
the real time dynamics of the monopole based on a point-like approximation.
Consequently, if there were long string networks with the monopoles in the
early universe, the monopole accelerates nearly to the speed of light emitting
electromagnetic radiations as a synchrotron accelerator, and collides to an
anti-monopole on the string. This collision event, which we call the
cosmological monopole collider, can produce much heavier particles than those
we can see today, e.g., at the Large Hadron Collider.
