How amazing was Hannibal's crossing of the Alps?

The Primary Record

Polybius was decidely UNimpressed:

Polybius, Book 3, Chapters 47-48
Some of the writers who have described this passage of the Alps, from the wish to impress their readers by the marvels they recount of these mountains, are betrayed into two vices most alien to true history; for they are compelled to make both false statements and statements which contradict each other.
While on the one hand introducing Hannibal as a commander of unequalled courage and foresight, they incontestably represent him to us as entirely wanting in prudence, and again, being unable to bring their series of falsehoods to any close or issue they introduce gods and the sons of gods into the sober history of the facts.
By representing the Alps as being so steep and rugged that not only horses and troops accompanied by elephants, but even active men on foot would have difficult in passing, and at the same time picturing to us the desolation of the country as being such, that unless some god or hero had met Hannibal and showed him the way, his whole army would have gone astray and perished utterly, they unquestionably fall into both the above vices.
For in the first place can we imagine a more imprudent general or a more incompetent leader than Hannibal would have been, if with so large an army under his command and all his hopes of ultimate success resting on it, he did not know the roads and the country? ...
Similarly, in what they say about the loneliness, and the extreme steepness and difficulty of the road, the falsehood is manifest. For they never took the trouble to learn that the Celts who live near the Rhone not on one or on two occasions only before Hannibal's arrival but often, and not at any remote date but quite recently, had crossed the Alps with large armies ... nor are they aware that there is a considerable population in the Alps themselves...
Of course Hannibal did not act as these writers describe, but conducted his plans with sound practical sense. He had ascertained by careful inquiry the richness of the country into which he proposed to descend and the aversion of the people to the Romans, and for the difficulties of the route he employed as guides and pioneers natives of the country, who were about to take part in his adventure.
On these points I can speak with some confidence as I have inquired about the circumstances from men present on the occasion and have personally inspected the country and made the passage of the Alps to learn for myself and see..

On the other hand, the fact that Polybius feels the need to attack these writers (plural) shows that - even in Polybius's day - Hannibal's crossing of the Alps had become legendary.  Many ancient writers believed (erroneously) that the Alps were impassable.  Meanwhile, Roman propaganda had every reason to enhance Hannibal's achievement; if you are fighting a superhuman opponent, it excuses why you lose the first few battles, and it makes your ultimate victory just so much more glorious ... so the Roman accounts bulled up Hannibal.

Livy is a case in point.  Although he has Hannibal berating his troops for their fear of the Alps - 'People actually live in the Alps, for goodness’ sake!' (30.7) - he throughout his account emphasises the huge dangers and difficulties Hannibal faced, and he even (erroneously) denies that the Great and Little St Bernard Passes were open in Hannibal's time (38.8).  So, again, the implication in Livy is that Hannibal's achievement was unprecedented.

The Secondary Interpretations

Most secondary sources seem prepared to echo the adulation of the ancients!

William Turner, 1812

When the great(est) British painter William Turner portrayed Hannibal Crossing the Alps, he portrayed tiny, powerless men, struggling against a landscape and weather of overwhelming power.  The only possible inference is that Hannibal's feat was a monument of achievement.

Modern historians
Similarly, it is impossible to read modern historians' accounts of the crossing without coming across words like 'legendary', 'remarkable', heroic' and 'amazing'. 

The wikipedia article on Hannibal's Crossing of the Alps comments:

Hannibal crossing the Alps, in 218 BC, was ... one of the most celebrated achievements of any military force in ancient warfare.

And the children's website socialstudiesforkids.com is even more effusive:

In a master stroke of military strategy that was as unconventional as it was daring, Hannibal made the courageous and outrageous decision to cross the mightly Alps with his invasion force

However, recently, Dexter Hoyos in Hannibal's Dynasty (2003) has criticised Hannibal on three grounds: 

a. he lost 36,000 men - hardly the actions of a 'successful' campaign;
b. he failed to negotiate a safe passage with the local tribes and had to fight his way through;
c. he mis-timed his march and ended up going over the pass in winter.