Complexity of languages

I am reading the dazzling book “through the language glass” by Guy Deutscher. The author states that, although it is possible to compare different languages on the base of the number of phonemes, the complexity of grammar and syntax, and the richness of the vocabulary, it is not possible to measure an overall index of language complexity, because it is not possible to compare, for instance, phonemes and syntax. It is not possible to compare phonemes and syntax, but the enthropy of phonemes and the enthropy of syntax can be compared. In ecology this is currently done: the number of a species in a habitat and the number of habitats in a region are very different things, but it is possible to calculate the enthropy of the former (alfa-diversity) and the latter (beta-diversity) and to sum these enthropy to obtain an overall measure of diversity of the landscape. Perhaps it is possible to sum the enthropy of phonemes, grammar, syntax and vocabulary and obtain a global measure of complexity.