Suspend courier showing obsessions (4-3)
I believe the answer is:
hang-ups
'obsessions' is the definition.
(hang-up can mean a neurosis or obsession)
'suspend courier' is the wordplay.
'suspend' becomes 'hang' (to hang is to suspend from something).
'courier' becomes 'UPS' (US delivery company).
'hang'+'ups'='HANG-UPS'
'showing' is the link.
(Other definitions for hang-ups that I've seen before include "In slang, emotional problems or inhibitions" , "Psychological problems" .)