After initially training as an apothecary, Koester enrolled at the Karlsruhe Academy to study painting under Carl Hoff and Claus Meyer. As a student he earned his livelihood with portrait commissions. Extensive travels and walks in the German and Austrian Alps were the source of a number of genre paintings. Apart from painting portraits, he dedicated himself more and more to genre painting and participated in numerous exhibitions. After finishing his studies, Koester moved to Klausen in the Tyrol. His new surroundings inspired him and led him to greater productivity, painting most of the time directly from nature. It was around this time that he first became interested in ducks, the subject that was to preoccupy him for the rest of his career. By using an impressionistic technique to capture the shimmering water, movement of the ducks, and contrast between light and shade, Koester produced paintings that proved to be extremely popular.

By the turn of the century, his work was beginning to be appreciated by an ever-widening audience, and in 1904 he received a gold medal at the World Fair in St Louis for the painting Enten. A further gold medal was awarded by the Prince Regent, Luitpold von Bayern for the painting Dem Ufer Zu.

After the war, Koester settled in Diessen on the Ammerseee. During this period he concentrated on still life paintings, but never abandoned the duck images completely.

Alexander Koester died on 21 December 1932 in Munich.