diff --git a/stdlib/LinearAlgebra/src/cholesky.jl b/stdlib/LinearAlgebra/src/cholesky.jl
index 60f171838c567..10aabb35928b8 100644
--- a/stdlib/LinearAlgebra/src/cholesky.jl
+++ b/stdlib/LinearAlgebra/src/cholesky.jl
@@ -181,12 +181,15 @@ end
 function _chol!(A::AbstractMatrix, ::Type{UpperTriangular})
     require_one_based_indexing(A)
     n = checksquare(A)
+    realdiag = eltype(A) <: Complex
     @inbounds begin
         for k = 1:n
+            Akk = realdiag ? real(A[k,k]) : A[k,k]
             for i = 1:k - 1
-                A[k,k] -= A[i,k]'A[i,k]
+                Akk -= realdiag ? abs2(A[i,k]) : A[i,k]'A[i,k]
             end
-            Akk, info = _chol!(A[k,k], UpperTriangular)
+            A[k,k] = Akk
+            Akk, info = _chol!(Akk, UpperTriangular)
             if info != 0
                 return UpperTriangular(A), info
             end
@@ -205,12 +208,15 @@ end
 function _chol!(A::AbstractMatrix, ::Type{LowerTriangular})
     require_one_based_indexing(A)
     n = checksquare(A)
+    realdiag = eltype(A) <: Complex
     @inbounds begin
         for k = 1:n
+            Akk = realdiag ? real(A[k,k]) : A[k,k]
             for i = 1:k - 1
-                A[k,k] -= A[k,i]*A[k,i]'
+                Akk -= realdiag ? abs2(A[k,i]) : A[k,i]*A[k,i]'
             end
-            Akk, info = _chol!(A[k,k], LowerTriangular)
+            A[k,k] = Akk
+            Akk, info = _chol!(Akk, LowerTriangular)
             if info != 0
                 return LowerTriangular(A), info
             end
diff --git a/stdlib/LinearAlgebra/test/cholesky.jl b/stdlib/LinearAlgebra/test/cholesky.jl
index 888ff065b5839..86a78a9e954fe 100644
--- a/stdlib/LinearAlgebra/test/cholesky.jl
+++ b/stdlib/LinearAlgebra/test/cholesky.jl
@@ -440,4 +440,11 @@ end
     end
 end
 
+@testset "issue #37356, diagonal elements of hermitian generic matrix" begin
+    B = Hermitian(hcat([one(BigFloat) + im]))
+    @test Matrix(cholesky(B)) ≈ B
+    C = Hermitian(hcat([one(BigFloat) + im]), :L)
+    @test Matrix(cholesky(C)) ≈ C
+end
+
 end # module TestCholesky