What happens to an unbaptized baby?
I'm surprised at you, Doc! You know that baptism has nothing to do with salvation:
From Romans 9: "Yet, before the twins were born or had done anything good or bad—in order that God's purpose in election might stand: not by works but by him who calls—she was told, ‘The older will serve the younger.’ Just as it is written: ‘Jacob I loved, but Esau I hated.’”
So for a baby, baptized or not, if it is predestined unto election, it enjoys eternal bliss with our Lord and Savior for all eternity. But if the baby is "a vessel of wrath prepared for destruction" (again, Romans 9), then it burns in hell for all eternity.
This is disturbing for some. It's disturbing for me! However, Paul does assert in Romans 9: "But who are you, O man, to talk back to God? “Shall what is formed say to him who formed it, ‘Why did you make me like this?’" 21 Does not the potter have the right to make out of the same lump of clay some pottery for noble purposes and some for common use?
An unbaptised baby does NOT go to hell or purgatory as they have not yet sinned of their own will. What happens to them? We say they go to “we don’t know” i.e. limbus infantium.