The complexity, predictability and predictive instability of the Western Mediterranean Oscillation index (WeMOi) at monthly scale, years 1856-2000, are analysed from the viewpoint of monofractal and multifractal theories. The complex physical mechanism is quantified by: (1) the Hurst exponent, H, of the rescaled range analysis; (2) correlation and embedding dimensions, mu* and d(E), together with Kolmogorov entropy, kappa, derived from the reconstruction theorem; and (3) the critical Holder exponent, alpha(o), the spectral width, W, and the asymmetry of the multifractal spectrum, f(alpha). The predictive instability is described by the Lyapunov exponents, lambda, and the Kaplan-Yorke dimension, D-KY, while the self-affine character is characterized by the Hausdorff exponent, H-a. Relationships between the exponent beta, which describes the dependence of the power spectrum S(f) on frequency f, and the Hurst and Hausdorff exponents suggest fractional Gaussian noise (fGn) as a right simulation of empiric WeMOi. Comparisons are made with monthly North-Atlantic Oscillation and Atlantic Multidecadal Oscillation indices. The analysis is complemented with an ARIMA(p,1,0) autoregressive process, which yields a more accurate prediction of WeMOi than that derived from fGn simulations.