I live with cross Alpine climber (4)
I believe the answer is:
ibex
'alpine climber' is the definition.
(the ibex is a mountain goat)
'i live with cross' is the wordplay.
'live' becomes 'be' (synonyms).
'with' means one lot of letters go next to another.
'cross' becomes 'x' (X is cross-shaped).
'i'+'be'+'x'='IBEX'
(Other definitions for ibex that I've seen before include "Mountain goat with curving horns" , "one used to high living" , "Pyrenean mountain goat" , "Wild goat of North Africa" , "Sure-footed type" .)