How do I get the value of an input element using angular.element('#id')

Explanation

You should use val method similar to jQuery's $.fn.val:

console.log(angular.element('#username').val());

Alternatively you can use value property of the pure HTMLInputELement:

console.log(angular.element('#username')[0].value);

... because angular.element instance is an array-like collection of HTMLElements with every element accessible by its index.

Correct approach

But... You should never read input value like this in context of Angular app. Instead, use ngModel directive and bind input value to angular model directly:

$scope.registerUser = function() {    
    console.log($scope.username);
};

where in HTML you have

<input type="text" ng-model="username">

This works for me

angular.element(document.getElementById('username')).val();