Postgres: Get value of a column corresponding to max of other column in a group

One option uses window functions in a subquery to rank the timestamps by increasing and descending value, then conditional aggregation in the outer query to bring the relevant values

select id, bucket,
    percentile_cont(0.5) within group (order by value) median_value,
    min(value) min_value, 
    max(timestamp_utc) filter(where rn_asc = 1) min_timestamp,
    max(value) max_value,
    max(timestamp_utc) filter(where rn_desc = 1) max_timestamp
from (
    select t.*, 
        row_number() over(partition by id, bucket order by value) rn_asc,
        row_number() over(partition by id, bucket order by value desc) rn_desc
    from (
        select t.*, time_bucket('60', timestamp_utc) as bucket 
        from rs.mytable t
        where 
            id in (1111,123)
            and timestamp_utc between '2020-11-05 10:00:15.748643'::timestamp 
                                  and '2020-11-05 16:35:48.750313'::timestamp
    ) t
) t
group by id, bucket 
order by id, bucket

Note that we need to compute the bucket first, and put it in the partition of the window function.