The characteristic function is $$\eqalign{ E\left[e^{itY}\right] &= E\left[ E\left[ e^{itY}|X \right]\right] \cr &= E \left[ \exp(it\mu X - t^2 X^2/2 \right] \cr &= \frac{\exp \left(\left(-(\alpha^2+\beta \mu^2) t^2 + 2 i \mu \alpha t\right)/\left(2 t^2 \beta + 2\right) \right)}{\sqrt {{t}^{2} \beta+1}}\cr}$$
It is certainly not a normal distribution, but might be approximated by a normal distribution when $\beta$ is small and $\alpha \ne 0$. In fact, by expanding this in a series in powers of $\beta$ and taking the inverse Fourier transform, I get a density
$$f(y) = \frac{\exp(-(y-\mu \alpha)^2/(2 \alpha^2))}{\sqrt{2 \pi |\alpha|}} \left( 1 + \frac { \left( 2\;{\alpha}^{4}+4\;x{\alpha}^{3}\mu+{x}^{2} \left( -5+{\mu}^{2} \right) {\alpha}^{2}-2\;{x}^{3}\alpha\mu+{x}^{4 } \right) }{2{\alpha}^{6}} \beta + \ldots\right) $$