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The answer is 'no'. The generic pair $A$ and $B$ will not have any nonzero linear combination that has a double eigenvalue. For a specific pair, take $$ A = \begin{pmatrix}-3&0&0&0\\0&-1&0&0\\0&0&1&0\\0&0&0&3\end{pmatrix} \quad \text{and}\quad B = \begin{pmatrix}0&0&0&3i\\0&0&i&0\\0&-i&0&0\\-3i&0&0&0\end{pmatrix}. $$ Then $$ \det(aA+bB - tI_4) = (t^2-a^2-b^2)(t^2-3a^2-3b^2), $$ and the roots of this polynomial in $t$ are distinct unless $a=b=0$. (Recall that we are assuming that $a$ and $b$ are real.)