iMac turns on automatically

A scheduled startup time may have been designated prior to upgrading to Ventura, a previously available Energy Saver feature Apple inexplicably removed from Settings.

Cancel it using the Terminal app. Please read Schedule your Mac to turn on or off in Terminal - Apple Support.

sudo pmset repeat cancel

... will cancel the current schedule. When typing that command you will be prompted for your login password. It will not be echoed, not even with •••• characters. Just type the password and press Return.

Hey Robert Blaize,

It would not be expected for the Mac to start up on its own when being shut down. There would not be a setting available to cause that to occur. We'd recommend reviewing this article to ensure the Mac is being fully shut down and not restarted or put into sleep mode: Shut down or restart your Mac - Apple Support

"Shut down your Mac

1. On your Mac, choose Apple menu  > Shut Down.
2. If you don’t want app windows that are open to reopen when your Mac restarts, deselect “Reopen windows when logging back in.”
3. A Mac is completely shut down when the screen is black. Some computers also don’t show an active power light or have any fan or drive noise.
4. Important: If you’re using a Mac laptop, wait for it to shut down completely before closing the display. If you close the display while the Mac is shutting down, it may not shut down properly and may start up more slowly the next time you turn it on. 

Always shut down your Mac before opening the case to install additional memory or other devices inside the computer. You may also need to shut down before connecting or unplugging certain peripheral devices. If you’re unsure, refer to the documentation that came with the device.

You don’t need to shut down your Mac when connecting or disconnecting cables, such as USB-C, Thunderbolt, or Ethernet.

Important: You can force a shutdown if necessary—for example, if your Mac becomes unresponsive—by pressing and holding the power button until your Mac shuts down. You may lose unsaved changes in open documents."

If that's not working as expected, contact Apple Support for further assistance here: Apple Support

Thanks!