I do not think that is useful as a measure of a distribution's normality. For large samples standard error of the median is defined as $1.253 \frac{\sigma}{\sqrt{n}}$, where standard error of the mean is $ \frac{\sigma}{\sqrt{n}}$, thus $SE_{median}=1.253SE_{mean}$ for large $n$. Both of these assume a normal distribution, and if the distribution is not normal, both estimates are misleading as measures of dispersion, yet maintain their intrinsic proportional relationship.

I think you may want to compare coefficient of variation $\frac{\sigma}{\mu}$ with interquartile ratio $\frac{IQR}{median}$, which only for a normal distribution would be $1.349\frac{\sigma}{\mu}=\frac{IQR}{median}$ or some variation thereof. For example, comparison of standard deviation and IQR forms the basis of a simple test for normality.

As per @Hugh, I do not think it is measure of a distribution's normality.

I think you may want to compare coefficient of variation $\frac{\sigma}{\mu}$ with interquartile ratio $\frac{IQR}{median}$, which only for a normal distribution would be $1.349\frac{\sigma}{\mu}=\frac{IQR}{median}$ or some variation thereof. For example, comparison of standard deviation and IQR forms the basis of a simple test for normality.