For Alpine road riding, I prefer a lightweight Audax style light tourer; you get low gears and mudguards for the wet days. Mostly I use my std tourer because I'm on tour hauling my panniers.
A rigid MTB or a more rugged hybrid with clearance for off-road tyres is fine for anything up to short red grade trails, you could probably get away with a CX bike with decent off-road rubber so long as you can get the saddle low enough that it doesn't crush your nuts with every impact. For fast technical descents, downhill parks, loose rutted trails and long off-road routes then suspension is a nice feature to look out for.
If you mean a good bike for climbing alpine passes on the road then the lighter the better.
I used to just ride my bike to work but now I find myself going out looking for bigger and bigger hills.
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If you mean a good bike for climbing alpine passes on the road then the lighter the better.