A novel class of complex valued kernel least mean square (CKLMS) algorithms is introduced with the aim to provide physical meaning to the mapping between the primal and dual space termed the independent CKLMS (iCKLMS). The general class of CKLMS algorithms is also extended in the widely linear sense to develop online kernel algorithms suitable for the processing of general complex valued signals, both circular and noncircular. The so-introduced augmented complex kernel least mean square (ACKLMS) algorithms are verified on adaptive prediction of nonlinear and nonstationary complex wind signals.