Recently, lattice-reduction-aided detectors have been proposed for multiple-input multiple-output (MIMO) systems to give performance with full diversity like maximum likelihood receiver, and yet with complexity similar to linear receivers. However, these lattice-reduction-aided detectors are based on the traditional LLL reduction algorithm that was originally introduced for reducing real lattice bases, in spite of the fact that the channel matrices are inherently complex-valued. In this paper, we introduce the complex LLL algorithm for direct application to reduce the basis of a complex lattice which is naturally defined by a complex-valued channel matrix. We prove that complex LLL reduction-aided detection can also achieve full diversity. Our analysis reveals that the new complex LLL algorithm can achieve a reduction in complexity of nearly 50% over the traditional LLL algorithm, and this is confirmed by simulation. It is noteworthy that the complex LLL algorithm aforementioned has nearly the same bit-error-rate performance as the traditional LLL algorithm.