Yeah, Mathematica can seem stubborn like that. Use FunctionExpand. E.g.,
In[17]:= Expand[(a + b + c + d)*
FunctionExpand[Conjugate[a - 3*b + c + d]]]
Out[17]= a Conjugate[a] + b Conjugate[a] + c Conjugate[a] +
d Conjugate[a] - 3 a Conjugate[b] - 3 b Conjugate[b] -
3 c Conjugate[b] - 3 d Conjugate[b] + a Conjugate[c] +
b Conjugate[c] + c Conjugate[c] + d Conjugate[c] + a Conjugate[d] +
b Conjugate[d] + c Conjugate[d] + d Conjugate[d]
Note that ComplexExpand is a bit different, it will write everything in term of real variables! You have to explicitly give it the list of variables that are complex (in your case a,b,c,d should all be assumed complex in principle)
In[9]:= ComplexExpand[(a + b + c + d)*Conjugate[a - 3*b + c + d], {a,
b, c, d}]
Out[9]= Im[a]^2 - 2 Im[a] Im[b] - 3 Im[b]^2 + 2 Im[a] Im[c] -
2 Im[b] Im[c] + Im[c]^2 + 2 Im[a] Im[d] - 2 Im[b] Im[d] +
2 Im[c] Im[d] + Im[d]^2 + Re[a]^2 - 2 Re[a] Re[b] - 3 Re[b]^2 +
2 Re[a] Re[c] - 2 Re[b] Re[c] + Re[c]^2 + 2 Re[a] Re[d] -
2 Re[b] Re[d] + 2 Re[c] Re[d] + Re[d]^2 +
I (4 Im[b] Re[a] - 4 Im[a] Re[b] - 4 Im[c] Re[b] - 4 Im[d] Re[b] +
4 Im[b] Re[c] + 4 Im[b] Re[d])
Good luck!