Is there an option in BackdropCMS core to notify an admin if a user creates a new account. In particular in cases when the setting for "Who can create a new account?" = "Visitors, but administrator approval is required."
It would appear, based upon limited research that Drupal 7 needs a contrib module to do this. I'm not seeing this module for BackdropCMS yet.
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I have the same problem and I see you are not getting much help here. You could create your own notification using a rule.
My thought also would be to use a rule.
Youre right this would need contrib, and there isnt any Backdrop port for this or similar yet. If you could make a port request that might help. See my suggestion here https://github.com/backdrop-ops/contrib/issues/266#issue-330860280 though.