In a cool damp climate, should I use hardwood or synthetic decking material for a deck that will be near a large tree?
EDIT

IPE is very very nice and really good material. As to your concern of slipping then using an anti slip decking oil will be most effective.   Be sure to look at DD5/DD6 Anti-slip specially designed for this use.
And protects from mould and algae



Sealed IPE



Please take a look at this post, maybe it could help you too.

How can I improve traction on an outdoor plywood ramp?

It is a bit of a shame putting anti slip strips that could ruin the look- but maybe wisely choosing areas for some strips is a good idea.

There are other solutions specially made for decks.

You can get these fancy strips that will plug into these pre made deck planks





You can also get various sizes and colours of antislip tack strips that you attach to any sort of decking plank



Also you can take a look at this alternative. It could be difficult to clean though. But if you deck has enough space for leaves and bit to fall through it- it might be goood

http://www.tacttiles.com/anti-slip/f3