Función MAX utilizada con la cláusula GROUP BY

Primero tienes que obtener el salario máximo para cada departamento:

SELECT department, max(salary) as max_salary
FROM test
GROUP BY department

luego puede volver a unir esta subconsulta a la tabla de prueba:

SELECT t.*
FROM
  test t INNER JOIN (
    SELECT department, max(salary) as max_salary
    FROM test
    GROUP BY department
  ) d ON t.department=d.department AND t.salary=d.max_salary