Using an Alias column in the where clause in Postgresql

You would need to either duplicate the case statement in the where clause, or my preference is to do something like the following:

SELECT *
FROM (
  SELECT 
      jobs.*, 
      (CASE WHEN lead_informations.state IS NOT NULL THEN lead_informations.state ELSE 'NEW' END) as lead_state
  FROM 
      "jobs"
      LEFT JOIN lead_informations ON lead_informations.job_id = jobs.id
      AND lead_informations.mechanic_id = 3
) q1
WHERE (lead_state = 'NEW')

I struggled on the same issue and "mysql syntax is non-standard" is not a valid argument in my opinion. PostgreSQL adds handy non-standard extensions as well, for example "INSERT ... RETURNING ..." to get auto ids after inserts. Also, repeating large queries is not an elegant solution.

However, I found the WITH statement very helpful (CTE's). It sort of creates a temporary view within the query which you can use like a usual table then. I'm not sure if I have rewritten your JOIN correctly, but in general it should work like this:

WITH jobs_refined AS (
    SELECT
        jobs.*,
        (SELECT CASE WHEN lead_informations.state IS NOT NULL THEN lead_informations.state ELSE 'NEW' END) AS lead_state
    FROM jobs
    LEFT JOIN lead_informations
        ON lead_informations.job_id = jobs.id
        AND lead_informations.mechanic_id = 3
)
SELECT *
FROM jobs_refined
WHERE lead_state = 'NEW'

MySQL's support is, as you experienced, non-standard. The correct way is to reprint the same expression used in the SELECT clause:

SELECT
    jobs.*, 
    CASE 
         WHEN lead_informations.state IS NOT NULL THEN lead_informations.state 
         ELSE 'NEW' 
    END AS lead_state
FROM
    jobs
    LEFT JOIN lead_informations ON
        lead_informations.job_id = jobs.id
        AND
        lead_informations.mechanic_id = 3
WHERE
    lead_informations.state IS NULL