MySQL 8.0

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CREATE TABLE YourTable
    (`StartDate` date, `StartTime` time, `EndDate` date, `EndTime` time, `status` varchar(7))
;
    
INSERT INTO YourTable
    (`StartDate`, `StartTime`, `EndDate`, `EndTime`, `Status`)
VALUES
    ('2020-03-02 00:00:00', '01:30:00', '2020-03-02 00:00:00', '09:00:00', 'running'),
    ('2020-03-04 00:00:00', '04:30:00', '2020-03-04 00:00:00', '09:00:00', 'running'),
    ('2020-03-04 00:00:00', '11:30:00', '2020-03-04 00:00:00', '19:30:00', 'running'),
    ('2020-03-05 00:00:00', '05:00:00', '2020-03-05 00:00:00', '11:15:00', 'running'),
    ('2020-03-05 00:00:00', '12:30:00', '2020-03-05 00:00:00', '22:08:00', 'running')
;
SELECT * FROM YourTable


StartDate
StartTime
EndDate
EndTime
status


2020-03-02
01:30:00
2020-03-02
09:00:00
running

2020-03-04
04:30:00
2020-03-04
09:00:00
running

2020-03-04
11:30:00
2020-03-04
19:30:00
running

2020-03-05
05:00:00
2020-03-05
11:15:00
running

2020-03-05
12:30:00
2020-03-05
22:08:00
running


WITH cte AS (
   -- using single datetime value for simpler logic
   SELECT * 
      , LAG (STR_TO_DATE(CONCAT(EndDate, ' ', EndTime), '%Y-%m-%d %H:%i:%s'), 1, NULL) 
            OVER (ORDER BY EndDate, EndTime) AS PrevEndDateTime
      , STR_TO_DATE(CONCAT(StartDate, ' ', StartTime), '%Y-%m-%d %H:%i:%s') AS StartDateTime            
   FROM   YourTable
)
SELECT CAST( DATE_FORMAT(COALESCE(PrevEndDateTime, StartDate),'%Y-%m-%d') AS DATE ) AS StartDate
       , CAST( DATE_FORMAT(COALESCE(PrevEndDateTime, StartDate),'%H:%i:%s') AS TIME ) AS StartTime
       , StartDate AS EndDate
       , StartTime AS EndTime
       , 'stopped' AS Status
FROM   cte
WHERE  StartDateTime <> PrevEndDateTime
OR     PrevEndDateTime IS NULL
;


StartDate
StartTime
EndDate
EndTime
Status


2020-03-02
00:00:00
2020-03-02
01:30:00
stopped

2020-03-02
09:00:00
2020-03-04
04:30:00
stopped

2020-03-04
09:00:00
2020-03-04
11:30:00
stopped

2020-03-04
19:30:00
2020-03-05
05:00:00
stopped

2020-03-05
11:15:00
2020-03-05
12:30:00
stopped

StartDate	StartTime	EndDate	EndTime	status
2020-03-02	01:30:00	2020-03-02	09:00:00	running
2020-03-04	04:30:00	2020-03-04	09:00:00	running
2020-03-04	11:30:00	2020-03-04	19:30:00	running
2020-03-05	05:00:00	2020-03-05	11:15:00	running
2020-03-05	12:30:00	2020-03-05	22:08:00	running

StartDate	StartTime	EndDate	EndTime	Status
2020-03-02	00:00:00	2020-03-02	01:30:00	stopped
2020-03-02	09:00:00	2020-03-04	04:30:00	stopped
2020-03-04	09:00:00	2020-03-04	11:30:00	stopped
2020-03-04	19:30:00	2020-03-05	05:00:00	stopped
2020-03-05	11:15:00	2020-03-05	12:30:00	stopped