Should I put my h-index on my CV?
A range of statistics can be useful in providing a quick snapshot of your research productivity. Common statistics include: number of refereed journal publications; number of publications that meet a criteria such as you being first author or the journal being above a certain impact factor or on a discipline specific list of quality journals; h-index; total amount of grant funding; etc. These would supplement a complete list of your publications.

With regards to h-index, make sure you make clear what citation database you are using.

I know that some people on this site object to the bureaucratic reductionism that can result from using measures like h-indices, impact factors and the like. Nonetheless, just because they are imperfect, does not prevent them from being useful. Such measures should not replace actually reading your work to assess its quality, but they can be useful in getting a rough handle on the impact and status of your research.

In terms of anecdotes, I have read several promotion applications that have successfully incorporated a range of such summary statistics to make the case for promotion. h-index, total grant funding, total publications, average teaching scores are all evidence, albeit imperfect that decision makers who are often outside your area will use to decide how to allocate resources like jobs, promotions, etc.