A simple way to do this is to add a new method to your scope.
$scope.startsWith = function (actual, expected) {
var lowerStr = (actual + "").toLowerCase();
return lowerStr.indexOf(expected.toLowerCase()) === 0;
}
Then change the filter syntax on your element.
<ul border="1px" ng-repeat="msg in messages | filter:search:startsWith">
Here is a working plunkr with the example above.
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