As I'm sure you know, the monthly cost will depend on a lot of factors like the size of the pool, the type of pump, the cost of water and electricity in your area, and whether or not you'll be heating the pool during winter. to get an accurate estimate, you would have to have all that information and give it to a pool professional.
With that said... It looks like the most common cost of pool maintenance is around $75 to $100 per month. This is if you do the maintenance yourself. If you choose to have a professional service do it, then you can probably expect the monthly cost to be closer to $200 or more.
Now keep in mind that I'm not a pool expert, so my numbers could potentially be completely off. My answer is based upon the information I got elsewhere on the internet. Here are some of the sources so you can see for yourself:
There are ways that you can save money on pool costs as well. One way is to make sure you have a variable speed pump. This type of pump can save you quite a bit every month on electric bills. Another is using a pool cover to keep the pool cleaner, warmer, and reduce evaporation.
I also found this website that spells out ways to reduce your monthly chemical costs. It's called the BBB method -- for bleach, baking soda, and borax as the chemicals to use instead of costly pool chemicals. They explain it better than I ever could so here it is: