The first place to start would be the sales_order_save_after event. This certainly will work, but will be called any time the order is updated and saved. Therefore the logic must consider when the order is newly created & complete straightaway, or when the order is marked as complete later on (the latter being the most common). You may need to adjust logic and acceptable end-state values for orders based on cancellations, multiple orders, etc.
/**
* Update customer attribute when order is completed.
*
* Need to catch two conditions:
* 1) Order is new AND `status` = complete
* 2) Order exists but `status` is changed to complete
*
* @param $obs Varien_Event_Observer
*/
public function adjustCustomerAfterComplete($obs)
{
/* @var $order Mage_Sales_Model_Order */
$order = $obs->getOrder();
if ($order->getStatus() === $order::STATE_COMPLETE
&& $order->getOrigData('status' !== $order::STATE_COMPLETE))
{
Mage::getModel('customer/customer')
->load($order->getCustomerId())
->setCustomAttr('new val') //custom attr code
->save();
//Another approach if you don't need events, etc.:
/*
$obj = new Varien_Object(
array(
'entity_id'=>$order->getCustomerId(),
'custom_attr'=>'new val'
)
);
Mage::getResourceModel('customer/customer')
->saveAttribute($obj,'custom_attr');
*/
}
}