Torns tab index, specifically when focus is on chat boxes, is one of the least logical I have come across.
For instance if I have both 'faction' and 'global' chat open, and my focus is on the textarea of global, (which is to the right of the two chat boxes) and I press tab, it goes directly to the user that sent the first message in the text box to the left, faction chat in this example, then the tab index proceeds to go down the names until reaching the textarea at the bottom of the text box.
From here the tab index goes to highlighting the URL. The text box to the most left of all text boxes will always react like this and the above sequence is always followed if the text box has another to the left of itself.
Ideally they should swap between the textareas, left to right, then looping back round to the first after reaching the last.