Solutions of the complex KdV equation and the complex KdV-
Burgers equation are studied theoretically and numerically. Attention is focused on whether their solutions are regular for all time. This is a difficult
issue partially because the conservation laws of the KdV equation no longer
yield a priori bounds for its complex-valued solutions in the $L^2$-space. The
problem is tackled here on several fronts including investigating how the regularity of the real part is related to that of the imaginary part, studying blow-up
of series solutions, and assessing the impact of dissipation. Systematic numerical simulations are performed to complement the theoretical results.