To answer the original question, there is a hidden-variable theory called Bohmian Mechanics, which agrees with the predictions of quantum mechanics (as far as they are unambiguous). Look it up on http://www.plato.stanford.edu .
As for the more general debate "is quantum mechanics complete and is this a matter for physics or philosophy?" I think that we can only say that the jury is still out on the subject. There are a growing number of physicists who think it is a matter of physics rather than philosophy. I am one of them, so let me explain why.
In the standard axiomatization of quantum mechanics, measurement plays a special role, but the theory makes no mention of what constitutes a measurement, i.e. what causes the projection postulate to apply rather than the usual unitary dynamics. This is known as the measurement problem. Now, for (almost) all practical purposes, we have no problem determining what a measurement is and the predictions of quantum mechanics agree well with experiment. However, modern experiments, such as those on SQUID rings, involve superpositions of macroscopically large systems and measuring devices which are well described by quantum mechanics. We can see that the distinction between 'quantum system' and 'measuring device' has become blurred.
Some approaches to the foundations of quantum theory make different predictions for the behaviour of such systems. For example, the 'spontaneous collapse' models predict a scale beyond which we cannot observe quantum coherences. Even Bohmian mechanics has a concept of 'non-equlibrium matter', which would behave differently from normal matter if it exists. We can see that these are actually different theories, with testable predictions, rather than mere 'interpretations', which is how they were originally seen. Since no theory invented by mankind has ever been the ultimate true theory of the universe, it is natural to assume that the same is true of quantum mechanics. To me it seems just as likely that new physics will emerge from ideas designed to deal with the conceptual difficulties of quantum mechanics as it is from other considerations.
Even interpretations that do not make any different predictions from quantum mechanics may prove to be quite important. For example, many of them, such as consistent histories, Bohmian mechanics and many-worlds, suggest radically different ways of constructing a theory of quantum gravity. If the consequences of these theories are ever tested, it may be possible to sort out the question of 'interpretation' by experiment.
Undoubtably there is interesting philosophy arising from
interpretations of quantum mechanics, but it is not at all clear where the dividing line between physics and philosophy actually lies. I believe that in the next few years we will see questions that are traditionally regarded as philosophical becoming accepted as real physical problems. This is just my opinion of course, but hopefully time will tell us the answer rather than debates on message boards.