At the end of the day, though, I can't help but think about Krzystof Kieslowski, another Polish filmmaker who, like Pawlikowski, started out in documentary and moved toward narrative film, spending some of his career making films outside his native country. Kieslowski's connection to the era in Ida would have been even more direct—he was born in 1941, while Pawlikowski was born in 1957. Kieslowski found himself often working as an outsider, trying to make work under the strictures of an oppressive regime while remaining Polish; Pawlikowski said in an interview that "I come from a family full of mysteries and contradictions and have lived in one sort of exile or another for most of my life. Questions of identity, family, blood, faith, belonging, and history have always been present."
Kiewslowski had a penchant for characters that are "doubled"—a sort of mysterious doppelganger, most clearly (though not solely) demonstrated in 1991's The Double Life of Veronique, in which a Polish girl and her French counterpart seem to be living lives along a dual track. Some read that film as representing the dual identities present in Europe at the time, setting up Communist Poland against the free West—one able to choose her future, and one still bound.
A similar doubling is happening in Ida, though Pawlikowski's isn't as blatant as Kieslowski's. Anna is Ida, Ida is Anna—they are the same person, quite literally, but also represent two possibilities for the same person. One reading of the ending of Ida might be that Anna has to try "Ida" on for size before knowing what choice she should make about her future. That's a reasonable conclusion, indicated early in the film.
But I think there's a second way to think about it: in the end, Ida realizes that she doesn't understand her aunt at all. She can't fathom why she made the life choices she did, and has to consider becoming her aunt's double, inhabiting her aunt's reality—wearing her clothes, engaging in her activities—in order to see what it was to be her, and to understand how she could have done what she did. And, perhaps, Ida's wondering if it's the best way that she can pray for her soul.
In a very imperfect way, this reminds me of the incarnation—the idea that God could inhabit a human body so that finite people could trust him to know what it is to be human, and to understand how Christ could intercede for them. To become another is in some way an act of love for that other person. Jesus knew what it was to be a person, and not one in an ivory tower.
On a night of unhappy drinking, Wanda tells Ida, "Your Jesus didn't hide out in a cave with books but went out into the world . . . This Jesus of yours adored people like me. Take Mary Magdalene."