If a series convergerges "absolutely" then, yes, it must converge. The other way is not true: if a seriies converges, it does not necessariily converge absolutely. For example, the series
converges but does not converge absolutely.
Most of the time, when you are dealing with complex series, you are dealing with power series. In that case, there always exist a "radius of convergence". Inside that radius, the power series muist converge absolutely, outside it, diverge. But on the radius of convergence, the series may converge absolutely, converge but not absolutely (converge "conditionallly"), or diverge.