Forcing complex output to take the form $a + b\,i$

Perhaps this?

ComplexExpand[Solve[z^2 == 1 + 2 I], TargetFunctions -> {Re, Im}]
(*
  {{z -> -5^(1/4) Cos[ArcTan[2]/2] - I 5^(1/4) Sin[ArcTan[2]/2]},
   {z ->  5^(1/4) Cos[ArcTan[2]/2] + I 5^(1/4) Sin[ArcTan[2]/2]}}
*)

Update: I saw Bill Watts' comment after I posted my first answer, which suggests FunctionExpand will help with the trig. functions. Simplifying the separate parts as follows gets the result closer to the desired form:

FunctionExpand@ComplexExpand[Solve[z^2 == 1 + 2 I]] /. 
 x_?NumericQ :> ToRadicals@FullSimplify[Re[x]] + I ToRadicals@FullSimplify[Im[x]]
(*
  {{z -> -I Sqrt[1/2 (-1 + Sqrt[5])] - Sqrt[1/2 (1 + Sqrt[5])]},
   {z ->  I Sqrt[1/2 (-1 + Sqrt[5])] + Sqrt[1/2 (1 + Sqrt[5])]}}
*)