Can I run 2 instances of the game on 2 different computers, 1 battle.net account or do I need an alternate battle.net account for each instance when on at the same time?
You can only purchase one copy of the game per Battle.net account.
As TheDarkJedi noted, you do need a second Battle.net account to simultaneously run a second copy of Diablo 3 (and also Diablo 4). Please take note that you cannot attach your second Battle.net account to the same phone number as your primary account as each account is only able to linked to a singular phone number that is not being used by any other Battle.net account. I just wanted to point this out so you know that you have to have a unique mobile phone number capable of receiving text messages (SMS) for account recovery purposes and for attaching an authenticator as a valid phone number is now also required when activating an authenticator so you have a backup recovery method in case you lose your authenticator’s serial and restore numbers (you should save a screenshot of these as well as a written/printed copy in a safe place to restore to new devices should you lose access to your current device).