Different behavior of blur event in different browsers

As you know, the issue is that different browsers choose to call event handlers in different orders. One solution is to give the other events a chance to fire by setting a timer for 0 milliseconds, and then checking the fields to see which (if any) is focused.

a.onfocus = function() {show(b);};

a.onblur = function() {
    setTimeout(function() {
        //if neither filed is focused
        if(document.activeElement !== b && document.activeElement !== a){
            hide(b);
        }
            }, 0);
};

//same action as for a
b.onblur = a.onblur;

Tested in Chrome, Firefox, Internet Explorer, and Safari. See full working example (edited version of your fiddle) at JSFiddle.net.

You can use an extravarible to check whether b is focused before hiding b. It worked in IE, Chrome & Firefox. I don;t have any other browser. You can check it.

var focusedB = false;
$("#a").focus(function(){
     $("#b").show();   
 });
 //if b is focused by pressing tab bar.
 $("#a").keydown(function(e){
     if(e.which === 9){
          focusedB = true;  
      }
   });
   $("#b").blur(function(){
        $("#b").hide();
   });
   $("#a").blur(function(){
       if(focusedB){
            focusedB = false;
        }else{
            $("#b").hide();
        }
    });
    $( "#b" ).mousedown(function() {
       focusedB = true;
    });