Edit in JSFiddle

$(function() {
    if (!window.localStorage) return;
    var value = window.localStorage.getItem('value');
    if (value) $('#field').val(value);    
});

$('#field').keyup(function() {
    if (!window.localStorage) return;
    window.localStorage.setItem('value', $('#field').val());
});

$(window).bind('storage', function(e) {
    var event = e.originalEvent;
    if (event.key == 'value') {
        $('#field').val(event.newValue);
    }
});
<input id="field">