Comic books and movies are loaded with Nazi super science and the roots go back all the way to mid-World War II comics. Robert A. Heinlein wrote a German lunar base story as early as 1947 and things have only grown more ridiculous.
Nazi moon bases and Nazi UFOs are also a long-standing staple of conspiracy theorists. I mean, how else would Hitler have flown into the lizard-man-haunted sanctuary of the hollow earth? But let's put all the comic book fun aside for a moment and consider the reality.
Science and War
The Nazis held power from 1933 until Germany's surrender in 1945. During this time, German scientists certainly pushed the boundaries of jet-and-rocket-powered propulsion. They secretly researched atomic weaponry and unleashed two of the more remarkable military technologies of the day: the dreaded V-weapons.
The V-1 was a pulse-jet powered cruise missile. The flying bomb claimed thousands of lives, but was ultimately susceptible to anti-aircraft defenses. But then came the V-2, the world's first long-range ballistic missile and the first man-made object to reach space. This supersonic weapon was virtually unstoppable by existent defensive measures, though it ultimately proved ineffective in turning the tide of the war.
The V-2 was a weapon of death and terror, but it also helped usher in the space age. The work of Wernher von Braun and other German rocket scientists fueled the efforts of both the United States and the Soviet Union. Von Braun himself became a driving force at NASA -- and may have helped fuel the myth of the Nazi space program in the process.
After the War
According to Smithsonian Space History Curator Michael Neufeld, during the Cold War, von Braun and key associates deliberately gave the false impressionism that while they'd been building weapons, they really only cared about space. Meanwhile, the Nazi power base itself had been devoted to total war and the only reason they supported rocketry at all was to develop better weapon technologies.
Make no mistake: Very little scientific activity took place under the Third Reich that did not directly benefit the war effort, and this was especially true of rocketry. As both Michael Neufeld and von Braun biographer Bob Ward related to me in a pair of 2011 interviews, a lot of the German rocket scientists supported weapons building and some were even enthusiastic Nazis. Von Braun himself was a right-wing nationalist German who had a lot of sympathy for the Nazis during the war.
So again, there was no Nazi space program. The Nazis certainly capitalized on a preexisting German rocketry fervor, but empowered it solely for war, not for exploration. But in this, the second world war and the cold war to follow certainly propelled space exploration.
As Neil de Grasse Tyson points out in his book "Space Chronicles," space exploration is a megaproject, much like he pyramids of old. Only three factors tend to propel such projects: war, economics and the praise of royalty or deity.
Humans are capable of such marvelous scientific achievements, but nothing speeds things along like our capacity to inflict brutal war.