why does the apple mouse continually disconnect and then reconnect. regardless of battery state - new or otherwise.

When did this start? I'm having the same problem for about 3 days now. I downloaded and applied the OS X El Capitan 10.11.4 Combo Update, ran Disk Utility and ran the First Aid on the drive. Haven't had a problem since (knock on wood as it's only been since mid morning today).

Bluetooth shares the 2.4 GHz ISM band with other household devices such as cordless telephones, wireless networks, baby monitors, and microwave ovens. If you see issues with your Bluetooth devices that you suspect might be due to frequency congestion, see AirPort and Bluetooth: Potential sources of wireless interference for further information.

Using a Bluetooth mouse, keyboard, or trackpad with your Mac - Apple Support

Update: it's back by maybe only once a day, i.e. lessened but not eliminated.

I had this issue and then found a feed from 2012 which discussed a solution. My mouse goes back to 2014, but this it stop the disconnecting, so worth a try.

Not all AA batteries are precisely the same size and the channel in the Apple products and the batteries themselves are on the wider side. As such average batteries move around in the mouse when moved, and every time they lose contact with battery plates than the bluetooth disconnected. Option 1 is to go to an Apple shop and buy apple branded rechargeable batteries. Option 2 is the one I opted for. Fold up a piece of paper about the size of a post-it note into a shape that fits between the 2 batteries. Then fold another one to the size that fits above the batteries to the width of 3-4mm wide. Close the mouse unit up. This reduces the movement of the batteries and should address the issue.

I hope this works.