I use a Polar H10, which is rated for swimming as it also has internal memory; you can tell it to start an activity from your phone, leave your phone behind, when you get back to it use it to stop the activity and download the data. Waterproof to 30m apparently.
The H10 does Bluetooth, ANT+ and gymlink - which was important to me as I use all three; the gymlink mainly to my ancient Polar HRM watch for use when out on the bike. It can connect to multiple things at once, including multiple simultaneous bluetooth. Does pretty much everything I want, but is a little spendy. And if you are a larger person , the included strap is a bit on the snug side. I'm a 44" chest, so not massive, and the strap is right at the end of its adjustment - though you can buy larger ones, which I have done for longer rides. You can't (or couldn't) buy it with the larger strap fitted instead of the smaller one.