Why get different results when solving the same complex equation using Reduce and Solve?

[mmamathes (Bill Blair)]

Reduce[{Norm[z1] == Norm[z2] == 2, z1 + z2 == Sqrt[3] + I,
y == Norm[z1 - z2]}, y, {z1, z2}]
Solve[{Norm[z1] == Norm[z2] == 2, z1 + z2 == Sqrt[3] + I,
y == Norm[z1 - z2]}] // FullSimplify

Why get different results when solving the same complex equation using Reduce and Solve?

[gianlucagorni]

I don’t know what happens here internally, but the problem goes away if we use Abs instead of Norm:

Reduce[{Abs[z1] == Abs[z2] == 2, z1 + z2 == Sqrt[3] + I,
  y == Abs[z1 - z2]}, {y, z1, z2}, Complexes]
Solve[{Abs[z1] == Abs[z2] == 2, z1 + z2 == Sqrt[3] + I,
  y == Abs[z1 - z2]}]