Eisenhower seeks and obtains both party's nominations either in 1948 or 1952
Extremely unlikely, though Ike himself supposedly once raised the possibility. Chester Bowles, one of the ADA liberals
http://spartacus-educational.com/USACada.htm who was trying to draft Eisenhower in 1948 (because Truman was sure to lose...) recalled in his memoirs:
"No one knew Eisenhower's political views, and I felt it was quite possible that he did not have any. I decided to find out. I called his office at Columbia University and an appointment was arranged for the following day. After a two-hour discussion I was convinced: (1) that he wanted to become President; (2) that this desire was qualified by his reluctance to participate in the turmoil of political life; (3) that his ideas on domestic policy were almost wholly unformed; and (4) that he was incredibly naive politically. As evidence of this latter point General Eisenhower seriously asked me at the end of our discussion if it might be possible for him to be nominated by both parties. I came away badly shaken."
IMO Eisenhower was not naive at all. He did not want to run for president in 1948, and was particularly annoyed by the way that Democratic leaders--ranging from ADA liberals to big city bosses to southern segregationists--were trying to get him to run to save the party from seemingly inevitable defeat, and he would have none of it. (He noted in his diary that the Republican demand for his candidacy came from grass roots Republicans, whereas the Democratic demand was from political leaders sure that Truman would lose. He didn't want either party's nomination.) I think he was just politely trying to give Bowles the brush-off.
As for 1952, remember that Ike only narrowly won the GOP nomination against Taft in OTL. If he had announced he would try to get the Democratic nomination as well, the Republicans would never have nominated him. As it was, his biggest struggle was to convince the party that he was a "real" Republican.