Here is a sample of the CSS, JavaScript and HTML I used to get the desired effects I wanted.  Please let me know where I can improve.
Note:  I used JavaScript to guarantee the link would open in a new window.  If you know a better way to do this. Please share.  It bugs me that I have anchor tags using onClick events.
<html>
<head>
<style type="text/css">
.bannerContainer
{
    position: relative;
    left: 0;
    top: 0;
    display:inline;
}
.myBanner
{
    position: relative;
    top: 0;
    left: 0;
}
.myBanner:hover + .newWindowButton
{
    display: inline;
}
.newWindowButton
{
    display: none;
}
.newWindowButton:hover
{
    display: inline;
}
.newWindowButton img
{
    /*This positions the maximize button.  You will have to play with the positioning to get it just right for your work.  This is what worked for me.*/
    position: absolute;
    top: -106px;
    left: 170px;
    width:30px;
    height:30px;
}
.pointer
{
    cursor: pointer;
}
</style>
<script type="text/javascript>
function myLinkClick(Url, isInNewLimitedWindow) {
      if (isInNewLimitedWindow) {
         window.open(environmentUrl, "_blank", "toolbar=no, scrollbars=no, menubar=no, resizable=yes");
      }
      else {
         window.open(environmentUrl);
      }
}
</script>
</head>
<body>
     <div class="bannerContainer">
          <a class="myBanner pointer" onclick="myLinkClick('http://myWebsite.com', false)">
              <img src="firstImage">
          </a>
          <a class="newWindowButton pointer" onclick="myLinkClick('http://myWebsite.com, true')">
              <img src="secondImage">
          </a>
     </div>
 </body>
</html>