Is there a normal 'combobox' out of the box? Currenty I use my own simple implementation using OutlinedTextField + DropDownMenu but it's far from perfect...

Don't think so, I also needed an input field with a list of options based on the current field value and I had to roll my own using drop down menu as well. That was a while back though, but I haven't heard anything about native support being added.

Have you seen

ExposedDropdownMenu

? https://cs.android.com/androidx/platform/frameworks/support/+/androidx-main:compose/material/material/samples/src/main/java/androidx/compose/material/samples/ExposedDropdownMenuSamples.kt