I put an air to air heat pump style hot water heater in my new house. I'd probably have gone with gas if that had been an option. One of the problems I had with it was if you used hot water for a shower in the morning, quite a common thing to do, the tank was heating after the cheap time of use pricing ended. So, less energy to heat the water, but could potentially cost the same as a regular tank getting the tank up to heat before the price went up. I never did the math, but you don't save as much as you'd think. I believe some of the newer ones can be programmed to avoid this issue, if you can wait until the evening for lots of hot water again.
Secondly, the principle is good in the summer as it pulls heat from the basement, where it was installed, to heat the water. In theory this should reduce the air con use. win-win. The problem is in the winter you are drawing heated air from the basement to heat your water. Since I spend a lot more time heating than I do cooling (only use the air con a few times a year in the worst heat/humidity) your efficiency for the house is not what you'd think.
I can't remember the brand, but there was an issue in a few years. A part corroded and the unit threw a code. As it turned out, there was also a burnt out backup element which I could have repaired myself, but the code made me call the repair guys. I wasn't present, but they quickly determined the burnt out element and replaced that, then determined that the other problem meant I'd need to spend almost as much as buying a new unit. I was already a few hundred dollars into the repair on a unit that was supposed to be saving me money. There was a secret warranty on the part, but labour was several hours, so I said forget it. I ran it on the backup elements as a regular heater and when it corroded out (I have really aggressive water) I went to Lowes and got a regular tank. Lesson learned.
Also, it's noisy as it has fans to pull the air through the heat exchangers.