When is a time series considered to be "complex"? (not complex-valued)
Most of the times one refers to a complex time-series to indicate that this is not, for example, a predictable periodic time-series or a sequence of repeating patterns with fixed seasonality. 

Complex time-series would be those coming e.g. from real-world measurements, and can be non-stationary, very noisy, have different and dynamic behaviors and seasonality, contain outliers and missing values, and so on.