The purpose, as it was later laid out in the Potsdam declaration and after, was to ensure the disappearance of Prussia as a factor in European politics. From the point of view of the allies (both Western and Soviet), Prussia had served primarily as a force of militarism and source of war in Central Europe, and its massive role in both World Wars within one generation laid the foundation for the notion that, since Prussia did not hold an ethnic identity like most of its neighbors, but was primarily a constitutional (and therefore largely philosophical) entity, it should be entirely dissolved. The notion that Prussia as an entity was the main culprit for everything that ailed Central European history had developed even before World War I, but was widespread by the end of WWII (I strongly recommend to read Christopher Clark, Iron Kingdom: The Rise and Downfall of Prussia, 1600-1947, pp. 670 ff., which lays out the case very clearly.)
That happened eventually in 1947, when Prussia was declared entirely dissolved as an international subject in the Four Power agreement. (This did not formally dissolve certain institutions of the Prussian state, which were handed over to its successors in the respective realm, but Prussia as a state ceased to exist in that moment by the stroke of a pen).
As capital of Prussia, Berlin was considered to be its functional and ideological centre. Hence, controlling Berlin meant to control Prussia, and to "cleanse" Berlin of Prussian militarism was therefore considered an ideological task for both Western and Soviet allied forces. Furthermore, Berlin had -- in the late 1930s -- contained almost 30% of Germany's industry, and therefore to control Berlin was too big a prize to be left to one ally alone.
The debate on what would happen post war between the allies were not stable or set in the mid 1940s. The main focus was to win the war first, defeat Germany soundly, occupy it entirely, and then figure out what to do with the country. Various suggestions and debates were made in this period, though I have not yet seen a comprehensive study myself of the mere plans and discussions around it in that period. There was of course the notorious Morgenthau plan, but also a confidential pondering of Churchill's (around '43 I believe) that Germany was to be divvied up into four or more separate countries and never to be reunited. In the end, all of these alternative plans became politically (and economically) impractical.

By separating Berlin into allied sectors, the allied powers could ensure a consistent weakening of the role of Berlin in German politics and economics (and thus its ascribed militarism) while keeping a close eye on the cultural liberalisation of Prussian culture (as the allies saw it). Clark points out correctly (pp. 675ff) that only the Soviet Union noticed the marked difference between the idea of Prussia (a constitutional law based state) and the ideology of Nazism, while the Western allies (in particular the British) tended to lump it all glaringly into one "militarist" basket.

As part of the joint allied control of Berlin, the various allied authorities began exorcizing what they saw as the roots of the authoritarian "Prussian Nazi" state. The Technical University of Berlin, for instance (formerly a "Technical College", not a university but more of a professional engineering school), was refounded as a university under the express condition that its engineers had to take mandatory classes in the humanities (i.e ethics and history)-- to ensure that never again Prussian engineers would engage in building machines and devices detached from knowing the moral consequences of their actions.

The entire role the Cold War eventually would play in the role of Berlin's politics really only began manifesting itself in the post war period after the basic decisions on Prussia were made. The "de-Prussianization of Prussia" happened to be continued while the Cold War went into full swing, and as the Cold War came to a head by the late 1940s, the original goal became secondary, but was still pursued by the allies into the 1960s. By that point, especially West Berlin had become such a "non-militarist" city that the need to "eradicate" Prussian militarism pretty much fell off the agenda without much fanfare. Nevertheless, since Berlin was formally not a part of either Germany but under the aegis of the allies, Berlin residents could not be drafted into any military unit on either side of the Iron curtain or the city itself (something the Soviets no longer cared about by 1960, but the policy was stuck to for West Berlin by the Western allies and West Germany all the way until 1990).

Given this perspective, the sectioning of Berlin was not just a gimmick. The allies (both East and West) considered it a vital part of formally occupying Germany together, and as part of the plan to exorcize the "Prussian militarist spirit" from the German soul. (Arguably, this has actually worked rather well, at least given the empirical observation that there is no revival movement for either Prussia or great military power politics in Germany today; that is, if one is to believe that it has a lot to do with Allied policies and not with the fact that Germans themselves, as a whole, were simply war-weary by 1945 anyway -- but that is a whole different enchilada).