Hi,
Most of the 23-pin to 15-pin adaptors simply make the relevant electrical connections between the relevant pins of each socket. I'd make a 95% sure guess that the adaptor you've got isn't a flicker fixer / scan doubler.
Your A4000 can be used with a PC monitor if you use the "Productivity" and "DoublePAL" / "DoubleNTSC" modes, but this will only work with system-friendly software (e.g. games/demos that bash the hardware directly will still open on the standard 15KHz modes).
Your choices are to either get a scan doubler / flicker fixer, which will be expensive and potentially difficult to locate (they appear from time to time at outrageous prices on eBay) or to get a replacement monitor for your Amiga that can sync down to the 15KHz modes.
Microvitec still makes 14" monitors that go up to 800x600 and sync to all the Amiga's 15KHz modes. The "Desk-Scan 1402" would do the job.
Alternatively, you might find a TFT screen that will sync to 15KHz modes, especially those that double as a TV or have composite video inputs in addition to VGA.
- Ali