I'm a GP and now work in emergency medicine. I have been qualified as a doctor for 24 years. I have worked in most specialties over the years, including psychiatry. I have also been a PPL since 1983.
If the copilot did indeed deliberately fly the aircraft into the Alps, as seems very likely, then that is at first sight completely baffling. If you just want to commit suicide, why on Earth would you want to take 150 innocent people with you? It's just not rational. And that's the thing -- this act was not rational, so therefore we have to look at what could plausibly make someone act in such an irrational manner.
In this case, it is actually very easy to see what could have made him act in such an irrational manner. We know he had a history of significant mental illness -- 18 months of what seems to have been fairly severe depression -- although we don't know the details. There has also been talk that he may have been in a situational crisis recently, precipitated by breaking up with his girlfriend. Against this background, it is entirely possible he could have developed either a psychotic depression or a brief reactive psychosis. Not all psychoses are accompanied by florid delusions and odd, thought-disordered behaviour. Some can be quite encapsulated and the person can appear normal in most respects:
Delusional disorder as a partial psychosis. - PubMed - NCBI
So if he had an encapsulated psychosis for whatever cause, it would be easy to see how he could believe, say as an example, (and I'm plucking contemporary iconography out of thin air to create a random, ludicrous proposition, which is how the delusions of people with psychosis are constructed) that in order to avoid a war between Russia and the West he had to crash his aircraft and kill himself and everyone on it because he and they were all agents of Vladimir Putin.
It's crazy and very unlikely, but this whole event is crazy and very unlikely and mad people can sometimes have crazy and very unlikely delusions (though obviously the vast majority of them are not any danger to the rest of the population) and we do know he had a history of mental illness.
I can't help remembering the engineer I saw in psychiatry as a medical student. He held down a responsible job, but he had an unshakeable, encapsulated delusion that his bosses had implanted a chip in his nose through which they kept an eye on him. In all other respects, he appeared totally plausible and functioned very well. It was quite an eye-opener and I've been thinking of him a lot in the past few days.
This was not a rational act and you really don't have to stretch that far to imagine a scenario whereby it could have occurred. The more difficult question is how you might prevent such a thing again and there is no clear answer to that, especially as we are now learning these things are not quite so unlikely as we once thought. Having two in the cockpit at all times is no panacea. A sharp push on the control column at 100 feet could have the same effect as a descent from 30,000 feet.
To pick up every pilot who might fly his aircraft into the ground, you would have to have a screening test so sensitive that it would be totally non-specific, which means that you would end up screening out tens of thousands of pilots, maybe even hundreds of thousands. And that's if you can even decide what the risk factors actually are. By the very nature of the action, the perpetrators are not around to be interviewed afterwards and it's quite possible, indeed likely, that each case has a set of unique, unpredictable precipitants and characteristics.