This is a strange story of mental illness (at least that's how I chose to interpret it). A man, Marc, gradually loses his grip on reality after shaving his mustache that was always there as part of his look as a man (or at least he thinks it was). The movie is seen through Marc's perspective which makes it difficult to discern reality from his bouts of schizophrenic hallucinations.
At some points the movie reminds me of Lost Highway in the manner with which it represented the split of the protagonist (or rather a metamorphosis) into two different characters. In La Moustache the split happens at the level of the "life" of the protagonist, his world is constantly subjected to transformations, while he more or less stays the same. However in LH we can discern the "true" part of the protagonist from his "imagined" part. That distinction is impossible in La Moustache ; we don't know where his madness starts and where it ends. He is a total mess, and it is upon this ambiguity that everything we see is built.
All in all, this movie was a really nice surprise that I highly enjoyed and that I recommend for fans of Lynch, Cronenberg and psychological thrillers/dramas.