This is not the same as to ask if Hitler had any Jewish ancestors. I do not think he had. The question is if he thought that he had, and to this question the answer seems to be positive. It looks like he at least seriously considered this to be possible.
Firstly, why I do not think he had Jewish ancestry? The only ancestor, who could be Jewish, is the father of Alois Hitler, Adolf’s father. Alois was born out of wedlock and her mother, Maria Schicklgruber, married Johann Georg Heidler when Alois was five years old. When Alois was ten, he was sent to live with Johann’s brother Nepomuk. Johann never acknowledged Alois as his son. Only after Johann’s death Alois decleared Johann as his father. Nazis investigated the issue and concluded that Hitler had no Jewish ancestors. Modern investigators also discard the theory of Jewish ancestors. The most likely candidate as the father of Alois is Johann Georg Heidler. The web has is one photo of Alois and two photos of Johann. In one of the two photos Johann looks quite similar to Alois. Some genetic connection is likely. My guess is that Johann was the father.
But this is beside the point. We are interested in whether Hitler thought his paternal grandfather was Jewish. There is evidence that he did. Hans Frank testified after the war that Hitler had told him in the 1930ies that one of his relatives tried to blackmail him by revealing his Jewish ancestry. Frank made for Hitler in investigation and reported the theory of Maria Schicklgruber having worked in the house of a Jew named Leopold Frankenberger at the time the child was born. Thus, Hitler had a good reason to believe that he might have a Jewish grandfather. He probably knew the family rumor, which the relative tried to use for blackmail. That would be natural, since his paternal grandfather had not acknowledged Alois, Adolf’s father, and it was bound to rise doubts of Alois’ father.
Furthermore, Hitler believed in racial purity and that pure Aryans were blond, tall and blue-eyed. Hitler’s eyes were dark blue, he was 175 cm tall and his hair was dark brown. That is not exactly the Aryan original type. Hitler thought that the French were mixed and that Germans also had mixture. Just from outer appearance he must have concluded that he himself had racial mixture with more southern peoples.
This seems to be true. Hitler’s Y-DNA haplogroup has been announced as E1b1b. This haplogroup characterizes old farmers from Northern Africa and Levant and it has spread to Southern Europe and even further, to Austria, but it is uncommon in Germany. Northern Europeans primarily derive from hunter-gatherers with Y-DNA haplogroups C, I, N. or to Indo-Europeans whose Y-DNA is R1. Thus, Hitler’s family was apparently not fully Northern European and there was admixture.
Hitler did not marry before the decision of making a suicide and he left no children. This is what he should have done assuming that he saw himself as a product of racial mixture, yet strongly condemning racial mixture. He did finally marry Eva Brown, but with no option for children. Eva Brown is an interesting case. Her mtDNA haplogroup, detected from a hair in her comb, seems to be N1b. That is characteristically Ashkenazi Jewish haplogroup. Thus, there is a good case that Eva Brown was partially Jewish. Her maternal grandmother was Josefa Winbauer. If is still not possible to track Josefa Winbauer further to verify if she was Jewish or not. She had a brother Alois Winbauer and her father was Winbauer. Nazis could not check if she had any Jewish ancestors, but they declared her Aryan. Thus, she was in a similar position as Hitler himself, but assuming that her mtDNA was N1b, she actually very probably was partially Jewish, unlike Hitler. Eva was mixed. Hitler probably though he himself was mixed. Such mixed people should stick together and not have children.
Hitler had Jewish friends in his youth, but at the end of the First World War he learned that Jews had betrayed Germany. After that he turned anti-Semitic and advocated racial purity. It seems that Hitler studied the Bible. The Bible clearly demands that Jews do not mix with other people. Hitler seems to have identified the original sin with racial mixture and the Jews with a snake. He wanted to complete what Jesus started. Jesus started the redemption. The redemption happens in the end of the times and it includes restoration of the Jews to Palestine. Hitler wanted to move the Jews out of Europe. He made a transfer agreement (Haavara) with Zionists facilitating immigration of German Jews to Palestine. He underlined a sentence from Paul de LaGarde’s book that Jews must be restored to Palestine. He did not work details on the Madagascar plan. He hardly intended to settle Jews to Eastern Europe as it still is Europe and they would return to Germany from so close distance. It seems to me clear that Hitler wanted to transfer the Jews to Palestine, not to any other place. He wanted to complete the redemption and to undo the original sin: racial mixture. When the Jews would be restored to Palestine, the thousand year reign of the Revelation would start.
What role can a mixed person, like what Hitler thought he was, play in this great redemption plan? It is the role of the messiah, who also is the Beast. The Jewish Messiah is a Jew but not a Jew. It means that the messiah is a mixture, a Benjamite: the tribe of Benjamin was mixed. Paul, when still called Saul, was a Benjamite. He persecuted the disciples of Jesus, Jesus being from the house of David (symbolically, actually symbolically or otherwise Hasmonean). King Saul was a Benjamite and he persecuted David from the tribe of Judah. Thus, the tribe of Judah, the Jews, must be persecuted at the end of the times by a Benjamite, a person with partial Jewish ancestry.
Hitler combined biblical racial purity with Darwinism: races, like species, should stay with their own. Racial mixture was against the highest natural law. This ideal could be achieved if every race would live in the place that was given to them. Thus, the Arians should take back their homeland, but also the Jews should be restored to the land assigned to them.
This all makes so perfect absurd sense that I am quite convinced that Hitler thought (incorrectly as it seems) that he was partially Jewish.