In his post https://www.bioconductor.org/packages/release/bioc/vignettes/phyloseq/inst/doc/phyloseq-FAQ.html#should-i-normalize-my-data-before-alpha-diversity-analysis Paul McMurdie mentions that normalizing prior to alpha diversity analysis is highly not recommended. However, it seems to be in the context of rarefying data, which from his seminal 2014 paper, is clearly inadmissible.
But what about other normalization methods like cumulative sum scaling (CSS) or transformation methods (e.g. variance-stabilizing) prior to estimating alpha diversity? Is it common to do this? I understand that certain alpha diversity indices have their own inherent normalization (e.g. Shannon uses relative abundances or total sum scaling (TSS)), but I'm not clear if this is commonly done or even sensible.