A new insight into the relationship between the complex circularity quotient and the coefficient of a linear minimum mean square estimator (LMMSE) which estimates a complex random variable from its complex conjugate is established. An adaptive version of this estimator, suitable for real time tracking of the degree of complex circularity is proposed based on the complex least mean square (CLMS) algorithm. For Gaussian data, the mean and mean square analyses show the effect of data non-circularity on the stability bounds of the real time circularity tracker. The concept is verified over simulations on both synthetic single- and multi-channel data and on real world wind signals with varying degrees of circularity.