My goals is to split a 9V battery into -4.5 V and +4.5 V.

First I will try out the circuit I found here. It works fine in Falstad:

When I put different resistors to the virtual ground the voltage do not stay 4.5 V though so I wonder how useful this circuit really is. I will test it out on an opamp.

Yes, worked fine connected to this circuit. Tested with 100 mVpp in @ 1 kHz. Amplification is x10.