  We study the complex-valued modified Korteweg-de Vries equation (mKdV) on the
circle. We first consider the real-valued setting and show global
well-posedness of the (usual) renormalized mKdV equation in the
Fourier-Lebesgue spaces. In the complex-valued setting, we observe that the
momentum plays an important role in the well-posedness theory. In particular,
we prove that the complex-valued mKdV equation is ill-posed in the sense of
non-existence of solutions when the momentum is infinite, in the spirit of the
work on the nonlinear Schr\"odinger equation by Guo-Oh (2018). This
non-existence result motivates the introduction of the second renormalized mKdV
equation, which we propose as the correct model in the complex-valued setting
outside of $H^\frac12(\mathbb{T})$. Furthermore, imposing a new notion of
finite momentum for the initial data, at low regularity, we show existence of
solutions to the complex-valued mKdV equation. In particular, we require an
energy estimate, from which conservation of momentum follows.
