ClearAll[expandPlusMinus]
expandPlusMinus = Flatten @* ReplaceAll[PlusMinus -> ({+##, # - #2} &)];
Examples:
2 ± Sqrt[3] // expandPlusMinus
{2 + Sqrt[3], 2 - Sqrt[3]}
(2 ± Sqrt[3]) ± 3 // expandPlusMinus
{5 + Sqrt[3], 5 - Sqrt[3], -1 + Sqrt[3], -1 - Sqrt[3]}
2 ± Sqrt[3] // expandPlusMinus // Map[PromptForm[x, #] &]
(2 ± Sqrt[3]) ± 3 // expandPlusMinus // Map[PromptForm[x, #] &]
Alternatively,
ClearAll[expandPlusMinus2]
expandPlusMinus2 = Block[{PlusMinus = Flatten@Through[{Plus, Subtract}@##] &}, #] &;
Examples:
2 ± Sqrt[3] // expandPlusMinus2
{2 + Sqrt[3], 2 - Sqrt[3]}
(2 ± Sqrt[3]) ± 3 // expandPlusMinus2
{5 + Sqrt[3], 5 - Sqrt[3], -1 + Sqrt[3], -1 - Sqrt[3]}