a*z^2 + b*(z - c)^2 == d

ComplexExpand[a*z^2 + b*(z - c)^2, {_}]

ComplexExpand[a*z^2 + b*(z - c)^2, {_}] /. a_ + Complex[0, 1] b_ :> Simplify[a] + I Simplify[b]
(*
 -Im[c]^2 Re[b] - Im[z]^2 (Re[a] + Re[b]) + Re[b] Re[c]^2 -
  2 Re[b] Re[c] Re[z] + Re[a] Re[z]^2 + Re[b] Re[z]^2 +
  2 Im[b] Im[c] (-Re[c] + Re[z]) +
  2 Im[z] (Im[c] Re[b] + Im[b] (Re[c] - Re[z]) - Im[a] Re[z]) +
 I (Im[b] (-Im[c]^2 + 2 Im[c] Im[z] - Im[z]^2 + (Re[c] - Re[z])^2) +
    Im[a] (-Im[z]^2 + Re[z]^2) + 2 (Im[c] Re[b] (Re[c] - Re[z]) +
    Im[z] (Re[a] Re[z] + Re[b] (-Re[c] + Re[z]))))
*)