Celebrating the pipe organ, the King of Instruments
Opinion is still divided as to whether Bach left Contrapunctus XIV unfinished - the four-part fugue that was to crown Art of Fugue. The long-standing assumption was that Bach died in the midst of composing it, but new theories have arisen in the last few decades to explain its abrupt ending.
One is that Contrapunctus XIV does not really belong to Art of Fugue at all because the principal theme is totally absent. It seems likely, however, that Bach intended to use the principal theme in the missing part of the fugue, and scholars have shown how Bach might have done this. 
Another opinion is that Bach probably did finish Contrapunctus XIV but the final manuscript was lost. This is based on differences between Bach’s first manuscript (completed around 1742) and the published edition of 1751. For example, the first edition contains pieces added since the 1742 manuscript (Contrapuncti IV and XIV) for which no original materials exist. [ 2] Did Bach prepare a final definitive manuscript before publishing Art of Fugue in the late 1740s?
A more recent view is that Bach intentionally left Contrapunctus XIV in an “unfinished” form as a musical puzzle - an invitation to other composers to either guess his intentions or come up with their own solutions. Bach and other organists often challenged each other to fugal contests, a sort of “dueling keyboards” in which one organist would begin a fugue and then suddenly hand off to the competitor, who continued and developed the theme. A number of composers have since tried their hand at the game (including Helmut Walcha, Lionel Rogg, and Michael Ferguson, as heard on Pipedreams No.2030).
 See Christoph Wolff, “The Last Fugue: Unfinished?”, in Current Musicology, No. 19 (1975), pp. 71-77.