First of all your check for a.FieldIDNumber = '1' should better be moved into WHERE clause:

SELECT

MySelectList

FROM SourceTable a

LEFT JOIN SourceTable a1

ON a.CompanyIdentifier = a1.CompanyIdentifier

AND a1.FieldIDNumber = '2'

LEFT JOIN SourceTable a2

ON

a.CompanyIdentifier = a2.CompanyIdentifier

AND a2.FieldIDNumber = '3'

LEFT JOIN SourceTable a3

ON

a.CompanyIdentifier = a3.CompanyIdentifier

AND a3.FieldIDNumber = '4'

LEFT JOIN SourceTable a4

ON

a.CompanyIdentifier = a4.CompanyIdentifier

AND a4.FieldIDNumber = '5'

WHERE a.FieldIDNumber = '1'

Looks like it is banking client data you are working with. Is your FieldIDNumber defined as varchar or int? If it's a numeric, then don't use quotes to avoid unnecessary data type conversion.

And the last one about your CXPACKET waits. That shows that you have problem with parallelism. Try to add OPTION (MAXDOP 1) after WHERE clause and see if it makes it run faster.