Wolfram Language knows about some simplifications for the product of the Gammafunction for complex conjugate numbers, i.e.
In[1]:= Gamma[n I] Gamma[-n I] // FullSimplify
Out[1]= (\[Pi] Csch[n \[Pi]])/n
In[2]:= Gamma[1 + n I] Gamma[1 - n I] // FullSimplify
Out[2]= n \[Pi] Csch[n \[Pi]]
In[3]:= Gamma[2 + n I] Gamma[2 - n I] // FullSimplify
Out[3]= n (1 + n^2) \[Pi] Csch[n \[Pi]]
In[4]:= Gamma[3 + n I] Gamma[3 - n I] // FullSimplify
Out[4]= n (4 + 5 n^2 + n^4) \[Pi] Csch[n \[Pi]]
In[5]:= Gamma[4 + n I] Gamma[4 - n I] // FullSimplify
Out[5]= Gamma[4 - I n] Gamma[4 + I n]
... but at some point it is stuck. With the help of WL I found these Identities for m=4,5 and 6.
In[8]:= N[Table[{Gamma[4 + n I] Gamma[
4 - n I] == (\[Pi] n (n^6 + 14 n^4 + 49 n^2 + 36))/
Sinh[n \[Pi]]}, {n, 1, 4}], 20]
Out[8]= {{True}, {True}, {True}, {True}}
In[9]:= N[
Table[{Gamma[5 + n I] Gamma[
5 - n I] == (\[Pi] n (n^8 + 30 n^6 + 273 n^4 + 820 n^2 + 576))/
Sinh[n \[Pi]]}, {n, 1, 4}], 20]
Out[9]= {{True}, {True}, {True}, {True}}
In[10]:= N[
Table[{Gamma[6 + n I] Gamma[
6 - n I] == (\[Pi] n (n^10 + 55 n^8 + 1023 n^6 + 7645 n^4 +
21076 n^2 + 14400))/Sinh[n \[Pi]]}, {n, 1, 4}], 20]
Out[10]= {{True}, {True}, {True}, {True}}
And of course more identities for other natural numbers can be found with some effort. Maybe someone can find a closed formula. for the case Gamma[m+I n]Gamma[m-I n], where m is a natural number.