why does my imac rechargeable mouse keeps disappearing

The following will require any USB keyboard and mouse. If you don’t have them, borrow or buy them, they are inexpensive!

1.     Connect the USB keyboard and mouse to the computer.

2.     Open System Preferences – Bluetooth

3.     Un-pair the wireless keyboard and mouse.

4.     Turn off Bluetooth

5.     Restart the computer in Safe Mode per the directions in How to use safe mode on your Mac - Apple Support

6.     Restart the computer normally.

7.     Open System Preferences – Bluetooth

8.     Turn on Bluetooth

9.     Re-Pair the wireless keyboard and mouse.

10.  Unplug the USB keyboard and mouse and begin testing the wireless devices again. 

The other issue with Blurtooth devices is radio interference. I have no issues with my Magic Mouse, Keyboard, and Trackpad unless I leave my iphone too close to the computer.

See: Resolve Wi-Fi and Bluetooth issues caused by wireless interference - Apple Support

Posting the same issue two times does not get a faster response. I assume the problem is that you do not ensure that the mouse is charged. One solution is to use a USB wired mouse instead. Then no problems.

Your welcome.