jQuery: Do I need both keyUp and change events?

onkeyup mostly fires when a key was pressed by the user and then released. onchange can happen if the user uses the browser's autocomplete feature. You should use the .on() method to catch both events, just in case:

$('input.searchbox').on('keyup change', function(e){
    $('input.txtfieldsearch').val($('input.searchbox').val());
    listViewUpdate();
    $('input.txtfieldsearch').trigger(e.type);
});

$('input.txtfieldsearch').on('keyup change', function(){
    $('input.searchbox').val($('input.txtfieldsearch').val());
    listViewUpdate();
});

You can use $.on() function and attach multiples handlers.

$("#element").on('keyup change', function (){
   // Your stuff...
});

You can use only "input" event if you don't need to provide support for the older browsers. It will be fired everytime a value changes in the input element. Here's a list of the supported browsers: https://developer.mozilla.org/en-US/docs/Web/Events/input