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])]}}
*)