Yup u can create your own notifications
While submitting the Service Request please set some text to one of the Sr type filed(sr type filed 40 = Accounting)
then write filter on SRM:Request once approved sent notification
sr type filed 40 = Accounting
notify hard code the people or while submitting set the er id's to one of the sr type filed
and make sr type filed 40 = $NULL$
You can also use process designer for sending notification if you are using it.
Process Designer can directly sends notifications that are defined in the process itself, as notification or email actions in Process Designer tasks.
Refer the below YouTube video link for basic understanding of Process Designer:
as mentioned by Sanjeeva, you can use the Notify Action in a Filter to send a custom Notification.
You could also extend this to use the Notification Engine as it is integrated to Service Request (and or Work Orders/Incident/Change).
Process Designer is also another option, without the need to create customisations - but is only applicable if you have it installed.
(Note: if you go down the Process Designer route, you cannot mix and match from the OOB PDT and Process Designer options.)