Suppose $A,B\in M_n(\mathbb C)$ are self-adjoint. Does there exist a constant $C>0$ depending only on $n$ such that $$ |A+iB| \leq C(|A| + |B|)? $$
One can take $C=1$ if $A$ and $B$ commute. More generally, if $A$ and $B$ are positive then this was answered in the affirmative in this question. The argument given is not obviously extendable to this situation.
The answer to the previous question goes on to show that this inequality can never hold in infinite dimensions.