Use an analytic function with a range window:

SELECT user_code,
       timestamp
FROM   (
  SELECT user_code,
         timestamp,
         COUNT(*) OVER (
           PARTITION BY user_code
           ORDER BY timestamp
           RANGE BETWEEN INTERVAL '0' HOUR PRECEDING
                     AND INTERVAL '1' HOUR FOLLOWING
         ) AS cnt
  FROM   table_name
)
WHERE  cnt >= 1000

Which, for the sample data:

CREATE TABLE table_name (user_code, timestamp) AS
SELECT 'Alice',
        TIMESTAMP '2024-01-01 00:00:00' + INTERVAL '3' SECOND * (LEVEL - 1)
FROM    DUAL
CONNECT BY LEVEL <= 1003
UNION ALL
SELECT 'Beryl',
        TIMESTAMP '2024-01-01 00:00:00' + INTERVAL '3' SECOND * (LEVEL - 1)
FROM    DUAL
CONNECT BY LEVEL <= 500
UNION ALL
SELECT 'Beryl',
        TIMESTAMP '2024-01-01 00:00:00' + INTERVAL '3' SECOND * (LEVEL + 501)
FROM    DUAL
CONNECT BY LEVEL <= 500
UNION ALL
SELECT 'Carol',
        TIMESTAMP '2024-01-01 00:00:00' + INTERVAL '3' SECOND * (LEVEL - 1)
FROM    DUAL
CONNECT BY LEVEL <= 900

Outputs the start of the hour ranges which contain 1000 or more rows:

If you want to return the 1-hour ranges then, from Oracle 12, you can use MATCH_RECOGNIZE:

SELECT *
FROM   table_name
       MATCH_RECOGNIZE(
         PARTITION BY user_code
         ORDER BY timestamp
         MEASURES
           FIRST(timestamp) AS timestamp_start,
           LAST(timestamp)  AS timestamp_end,
           COUNT(timestamp) AS cnt
         PATTERN (start_row within_1_hour+)
         DEFINE
           within_1_hour AS timestamp <= FIRST(timestamp) + INTERVAL '1' HOUR
       )
WHERE  cnt >= 1000

Which, for the sample data, outputs:

fiddle

Maybe this could help you:

--    S a m p l e    D a t a : 
CREATE TABLE tbl (user_code, timestamp) AS
SELECT 'Alice', TIMESTAMP '2024-01-01 00:00:00' + INTERVAL '20' MINUTE * (LEVEL - 1)
FROM    DUAL  CONNECT BY LEVEL <= 9
UNION ALL
SELECT 'Beryl', TIMESTAMP '2024-01-01 00:00:00' + INTERVAL '20' MINUTE * (LEVEL - 1)
FROM    DUAL  CONNECT BY LEVEL <= 4
UNION ALL
SELECT 'Beryl', TIMESTAMP '2024-01-01 00:00:00' + INTERVAL '20' MINUTE * (LEVEL + 1)
FROM    DUAL  CONNECT BY LEVEL <= 3
UNION ALL
SELECT 'Carol', TIMESTAMP '2024-01-01 00:00:00' + INTERVAL '20' MINUTE * (LEVEL - 1)
FROM    DUAL CONNECT BY LEVEL <= 2

I need to find out if any user has inserted 1000 or more records within a 1 hour time window on a specific date

Create a CTE (grid) to get a date and timeframes of 1 hour before and 1 hour after the current row. Then count previous and following rows per user and date …

WITH
  grid AS 
    ( SELECT   user_code, timestamp, a_date, prev_hour, next_hour,
               Count(*) Over(Partition By user_code, a_date 
                             Order By timestamp
                             Rows Between Unbounded Preceding And Current Row) as cnt_prev, 
               Count(*) Over(Partition By user_code, a_date
                             Order By timestamp
                             Rows Between Current Row And Unbounded Following) as cnt_next 
      FROM   ( Select   user_code, timestamp, Trunc(timestamp) as a_date,
                        timestamp - INTERVAL '1' HOUR as prev_hour,
                        timestamp + INTERVAL '1' HOUR as next_hour
               From     tbl
             )
      WHERE  timestamp Between prev_hour And timestamp 
           OR
             timestamp Between timestamp And next_hour
    )

… now get Max counts bigger than some value (here it is 8 – you put 1000) and group by columns that you want to (here it is user_code, a_date and a_time – that could be just the user_code if you want)

--      M a i n    S Q L :
Select   user_code, a_date, To_Char(timestamp, 'hh24:mi:ss') as a_time, Max(cnt_prev) as cnt_prev, Max(cnt_next) as cnt_next
From     grid
Group By user_code, a_date, To_Char(timestamp, 'hh24:mi:ss')
HAVING   Max(cnt_prev) > 8 OR Max(cnt_next) > 8
Order By user_code, a_date, To_Char(timestamp, 'hh24:mi:ss')

R e s u l t :

fiddle