結果/ID に基づいて最終結果を計算する

うーん。これは優先順位付けのクエリです。 row_number() のように聞こえます

select t1.personid, t1.employment
from (select t1.*,
             row_number() over (partition by personid
                                order by (case when employment <> 'Stayed the same' then 1 else 2 end),
                                         date desc
                               ) as seqnum
      from t1
     ) t1
where seqnum = 1;