I suppose you could attach the value to the jqXHR object in the error callback, so long as jQuery passes the same object to both callbacks:
error: function(jqXHR, textStatus, errorThrown) {
jqXHR.errorThrown = errorThrown;
},
complete: function(jqXHR, textStatus) {
console.log(jqXHR.errorThrown);
}
fiddle demonstrating the general concept: http://jsfiddle.net/9kQna/