Postgres - rows to columns

This is not what you request, but maybe be useful for you. Last column is an array with the structure: {'course_name: course_completed', ...}.

SELECT u.firstname,u.lastname, u.institution as department, u.department as company
    array_agg(c.shortname || ': ' || to_char(to_timestamp(p.timecompleted)::date,'YYYY-MM-DD')) AS courses_completed
FROM mdl_course_completions AS p
JOIN mdl_course AS c ON p.course = c.id
JOIN mdl_user AS u ON p.userid = u.id
WHERE c.enablecompletion = 1 AND u.firstname is NOT NULL AND p.timecompleted is NOT NULL
GROUP BY u.firstname,u.lastname, u.institution, u.department
ORDER BY u.firstname

output will be:

firstname   lastname   department   company   courses_completed
u1          u1         x            x         {"c1: date", "c2: date"}
u2          u2         x            x         {"c1: date", "c2: date"}

Other option is to use crosstab function (like you said): Postgresql crosstab documentation. You have to install tablefunc module to use crosstab. Read the documentation and if you think it could be useful for you I could help you to write the query (but I will need to know the tables structure and some example data).

You can use conditional aggregate, for example by using the FILTER clause:

SELECT
    firstname, lastname, department, company,
    MAX(completed) FILTER (WHERE course = 'c1') as c1,
    MAX(completed) FILTER (WHERE course = 'c2') as c2
FROM 
    -- <joins>
GROUP BY firstname, lastname, department, company

You need to group by the columns which are part of the same group. Then you can execute any aggregation function you like on the columns you want to pivot. With the appended FILTER clause you can filter all records which should be recognized for the aggregate.