Fascinated by an unusual demeanour (9)
I believe the answer is:
enamoured
'fascinated' is the definition.
('enamour' can be a synonym of 'fascinate')
'an unusual demeanour' is the wordplay.
'an unusual' is an anagram indicator.
'demeanour' with letters rearranged gives 'ENAMOURED'.
'by' is the link.
(Other definitions for enamoured that I've seen before include "Very fond of" , "Inspired with love" , "Charmed" , "Besotted" , "Smitten" .)