This should do it (after my 3rd edit :-) )
It's quite verbose, but pretty easy to see what it's doing. I'm sure you can shorten it:
Here is a working example: http://jsfiddle.net/59SPw/
html:
<select id="my_element_id">
<option>Alex</option>
<option selected="selected">IS</option>
<option>Cool</option>
</select>
Js:
var selectedIndex = $('#my_element_id :selected').index();
if(selectedIndex > -1)
{
selectedIndex = selectedIndex + 1; //index() is 0 based, nth-child is 1 based
var prevIndex = selectedIndex - 1;
var nextIndex = selectedIndex + 1;
var prev = $('#my_element_id :nth-child(' + prevIndex + ')').val();
var next= $('#my_element_id :nth-child('+ nextIndex + ')').val();
alert(prev || "no previous");
alert(next || "no next");
}