So, after some more digging this morning and banging my head I found a solution using SPServices. First, convert the field from a complex to a simple drop down:
$().SPServices.SPComplexToSimpleDropdown({
columnName: "myColumn"
});
Then, you can get the ID (and then the value) by using what SPServices has cleverly injected:
$('#SPComplexToSimpleDropdown_myColumn').val();
I'm still not sure of a way to do this without SPServices but I think it has to do with how SharePoint is doing funny business with the input
field.
#PS: set the value
PS: set the value
If you want to set the value of the dropdown you need to fire the change()
event after setting it so the internal code gets tapped.
$('#SPComplexToSimpleDropdown_myColumn').val(15).change();