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.
2. Prefer Inner Join Over Left Join (modified query below): We identified that one or more left joined entities (e.g. `click_tracking`) are used in the 'where' clause, in a way that allows to replace it with an optimized inner join. Inner joins can be fully optimized by the database, while Left joins apply limitations on the database's optimizer.

Optimal indexes for this query:

ALTER TABLE `click_tracking` ADD INDEX `click_tracking_idx_event_name_user_email_timestamp` (`event_name`,`user_email`,`timestamp`);
ALTER TABLE `click_tracking` ADD INDEX `click_tracking_idx_user_email` (`user_email`);

The optimized query:

SELECT
        t1.user_email,
        t1.event_name,
        t1.timestamp,
        t2.event_name,
        t2.timestamp 
    FROM
        click_tracking t1 
    INNER JOIN
        click_tracking t2 
            ON t1.user_email = t2.user_email 
    WHERE
        t1.event_name = 'Started' 
        AND t2.event_name = 'Completed' 
        AND t1.timestamp <= '2021-05-17' 
        AND t1.timestamp >= '2021-05-16' 
        AND t2.timestamp <= '2021-05-17' 
        AND t2.timestamp >= '2021-05-16' 
    GROUP BY
        t1.user_email 
    ORDER BY
        t1.user_email ASC