Can the error function be expressed in terms of other special functions?

There are many identities reminding one in the question (unclear what kind of relation has been intended), e.g. expressing FresnelS in terms of Erfi

FullSimplify[-1/4 (1 + I)( Erfi[(1 + I)Sqrt[Pi]z/2]- I Erfi[(1 - I)Sqrt[Pi]z/2])]

FresnelS[z]

Let's demonstrate other relations:

FullSimplify[{-1/4 (1+I)(I  Erfi[(1+I)Sqrt[Pi]z/2]- Erfi[(1-I)Sqrt[Pi]z/2]),
              -(Sqrt[-z^2]/z) - (z/Sqrt[Pi]) ExpIntegralE[1/2, -z^2],
              (z/Sqrt[-z^2])(1 - (E^z^2/Sqrt[Pi])HypergeometricU[1/2, 1/2, -z^2])
              }]//Column

  FresnelC[z]
  Erfi[z]
  Erfi[z]}

Such identities can be found exploiting MathematicalFunctionData and MathematicalFunction (the latter new in version 12), nontheless one can start with

MeijerGReduce[Erfi[x],x]

 Entity["MathematicalFunction","Erfi"]["AlternativeRepresentations"]

as well as

Entity["MathematicalFunction","Erfi"]["HypergeometricRepresentations"]

A convenient way of exploring mathematical data involves Manipulate, e.g.

Manipulate[ Entity["MathematicalFunction","Erfi"][z], 
            {z, Entity["MathematicalFunction","Erfi"]["Properties"]}]

various items of the following can yield other identities:

MathematicalFunctionData["Properties"]

U = Erfi[((1/2 + I/2) (R - z))/Sqrt[k R]] + 
   Erfi[((1/2 + I/2) (R - Sqrt[D^2 + z^2]))/Sqrt[k R]];

Use ComplexityFunction to penalize the use of Erfi

U2 = FullSimplify[U, 
  ComplexityFunction -> (LeafCount[#] + 
      1000 Count[#, _Erfi, {0, Infinity}] &)]

(* (1 + I) (FresnelC[(R - z)/(Sqrt[π] Sqrt[k R])] + 
   FresnelC[(R - Sqrt[D^2 + z^2])/(Sqrt[π] Sqrt[k R])] + 
   I (FresnelS[(R - z)/(Sqrt[π] Sqrt[k R])] + 
      FresnelS[(R - Sqrt[D^2 + z^2])/(Sqrt[π] Sqrt[k R])])) *)

The expressions are equivalent

U == U2 // FullSimplify

(* True *)

However, you now have four special functions rather than two.