The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.

Optimal indexes for this query:

CREATE INDEX example_idx_user_name_tm ON "example"."example" ("user_name","tm");

The optimized query:

SELECT
        t.user_name,
        t.place_name AS r1_place,
        max(t.tm) AS r1_tm,
        t2.place_name AS r2_place,
        min(t2.tm) AS r2_tm 
    FROM
        example.example AS t 
    JOIN
        example.example AS t2 
            ON t.user_name = t2.user_name 
            AND t.tm < t2.tm 
            AND t.place_name <> t2.place_name 
    WHERE
        t.tm BETWEEN '2020-02-25 00:00:00' AND '2020-03-25 15:00:00' 
        AND t2.tm BETWEEN '2020-02-25 00:00:00' AND '2020-03-25 15:00:00' 
    GROUP BY
        t.user_name,
        t.place_name,
        t2.place_name