Really great question!
I have not seen a lot of stores ask about that, but I think it depends on the nature of your store and possibly customers. If you have a store that remains fairly static with prices, then this may not be worth it.
If you have a store that has prices that will often fluctuate, or you have a store that often places products on sale, then I think there is room to send price alerts for sure!
The best app I found on this is Followprice: https://apps.shopify.com/followprice
This would allow customers to track products and receive price alerts. You could also look to send alerts using a newsletter for those that sign up to receive your newsletter.
You can also send discount codes in those newsletters to hopefully gather even more traffic.
Lots of options and possibilities with this I think, which is great! Just depends on the nature of your store and what your customers expect.
I hope that works out for you and your store, if you have any further questions about this or anything Shopify related, don't hesitate to reach out to us, we are happy to help.