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. Replace Join With Exists To Avoid Redundant Grouping (modified query below): When a joined table isn’t used anywhere other than in the WHERE clause, it's equivalent to an EXISTS subquery, which often performs better. In cases where the DISTINCT or GROUP BY clause contains only columns from the Primary key, they can be removed to further improve performance, as after this transformation, they are redundant.

Optimal indexes for this query:

CREATE INDEX jobs_idx_viewable_timestamp ON jobs (viewable,timestamp);
CREATE INDEX processes_idx_group_id ON processes (group,id);
CREATE INDEX status_idx_id_name ON status (id,name);
CREATE INDEX tasks_idx_id_processid ON tasks (id,processid);

The optimized query:

SELECT
        DISTINCT TOP 500 j.correlationid,
        j.timestamp 
    FROM
        jobs AS j 
    WHERE
        (
            j.viewable = 1 
            AND 1 = 1 
            AND 1 = 1
        ) 
        AND (
            EXISTS (
                SELECT
                    1 
                FROM
                    status AS s 
                INNER JOIN
                    tasks AS t 
                INNER JOIN
                    processes AS p 
                WHERE
                    (
                        (
                            (
                                (
                                    j.statusid = s.id
                                ) 
                                AND (
                                    s.name IN (
                                        'Dead', 'Process Complete'
                                    )
                                )
                            ) 
                            AND (
                                j.taskid = t.id
                            )
                        ) 
                        AND (
                            t.processid = p.id
                        )
                    ) 
                    AND (
                        p.[group] = 'SapDocument'
                    )
            )
        ) 
    ORDER BY
        j.timestamp DESC