In Schwartz "Quantum field theory and the standard model" pag 160, the generators of the rotation are Hermitian, while the generators of boosts are anti-Hermitian, as an example:
$ J_1 = \left( \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -i \\ 0 & 0 & i & 0 \\ \end{matrix}\right)\,, K_1 = \left( \begin{matrix} 0 & -i & 0 & 0 \\ -i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{matrix}\right) $
So you have $\langle \psi | \psi \rangle $ is rotation invariant since $e^{i\theta J_1}$ is Unitary an then
$\langle \psi | \psi \rangle \to \langle \psi | \left(e^{i\theta J_1}\right)^\dagger e^{i\theta J_1} |\psi \rangle = \langle \psi | e^{-i\theta J_1} e^{i\theta J_1} | \psi \rangle = \langle \psi | \psi \rangle $
But for the boost you have that $\left(e^{i\beta K_1}\right)^\dagger = e^{i\beta K_1} $.
So why generator of boosts are chosen to be anti-Hermitian instead of Hermitian?
Furthermore, using Hermitian generators for rotation you get the bracket $[J_i, J_j] = i\epsilon_{ijk}J_k$, so in this way the $i$ factor makes the Lie algebra $\mathfrak so(1,3)$ not closed in respect to the brackets, unless you re-define the bracket with a $-i$ factor.