The optimization process and recommendations:

1. Avoid Subqueries (query line: 17): We advise against using subqueries as they are not optimized well by the optimizer. Therefore, it's recommended to join a newly created temporary table that holds the data, which also includes the relevant search index.
2. 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.
3. Explicitly ORDER BY After GROUP BY (modified query below): By default, the database sorts all 'GROUP BY col1, col2, ...' queries as if you specified 'ORDER BY col1, col2, ...' in the query as well. If a query includes a GROUP BY clause but you want to avoid the overhead of sorting the result, you can suppress sorting by specifying 'ORDER BY NULL'.

Optimal indexes for this query:

ALTER TABLE `agentstatusinformation` ADD INDEX `agentstatusinforma_idx_userid_pkey` (`userid`,`pkey`);
ALTER TABLE `axpuser` ADD INDEX `axpuser_idx_pkey` (`pkey`);

The optimized query:

SELECT
        DISTINCT (SELECT
            axpuser.loginid 
        FROM
            axpuser 
        WHERE
            axpuser.pkey = age.userid),
        CASE 
            WHEN age.agentstate = 1 THEN 'Ready' 
            WHEN age.agentstate = 3 THEN 'Pause' END AS state 
FROM
agentstatusinformation age 
WHERE
(
    age.userid, age.pkey
) IN (
    SELECT
        agentstatusinformation.userid,
        max(agentstatusinformation.pkey) 
    FROM
        agentstatusinformation 
    GROUP BY
        agentstatusinformation.userid 
    ORDER BY
        NULL
)