I found a simple answer which works for me and I'd overlooked. I just switched the positioning to absolute in the CSS class for the AutoCompleteExtender and then set the Z-Index for it. The suggestion list for the autocomplete now appears above all other elements.
Code for control in .aspx
I have applied my own CSS:
<cc1:AutoCompleteExtender ID="componentID_AutoCompleteExtender" runat="server"
TargetControlID="componentID"
ServicePath="ImageComponentService.asmx"
ServiceMethod="GetComponentMatches"
MinimumPrefixLength="3"
CompletionInterval="1000"
EnableCaching="true"
CompletionSetCount="10"
CompletionListCssClass="CompletionListCssClass"
CompletionListItemCssClass="CompletionListItemCssClass"
CompletionListHighlightedItemCssClass="CompletionListHighlightedItemCssClass"
OnClientItemSelected="itemSelected"
Enabled="true" FirstRowSelected="true"
BehaviorID="AutoCompleteEx">
</cc1:AutoCompleteExtender>
CSS
.CompletionListCssClass
{
font-size: 11px;
color: #000;
padding: 3px 5px;
border: 1px solid #999;
background: #fff;
width: 300px;
float: left;
z-index: 1;
position:absolute;
margin-left:0px;
}