While Nadarajah (2005) may have used the term 'generalised Normal' to describe a density that nests this form, there are more suitable names that extend far further back in time, and which accordingly seem much more appropriate. 

In particular, I believe this should properly referred to as a **Subbotin distribution** (Subbotin 1923). Other later references include: 

* Diananda (1949) 

* Turner (1960) 

* Zeckhauser and Thompson (1970) 

* McDonald and Newey (1988)

* Mineo and Ruggieri (2005)

The functional form given by Subbotin (1923) defines the pdf as:

$$f(x) = \frac{\alpha }{2 b \Gamma \left(\frac{1}{\alpha }\right)}\text{exp}\left[-\left|\frac{x}{b}\right|^{\alpha }\right]$$

Subbotin used parameter $b = 1/h$, but the functional form is otherwise identical to that given here. Here is a plot of the pdf with $b=2$, as parameter $\alpha$ varies:


<img src="http://www.tri.org.au/se/Subbotinpdfplot.png">

In this form: $$Var(X) = \frac{b^2 \Gamma \left(\frac{3}{\alpha}\right)}{\Gamma \left(\frac{1}{\alpha}\right)}$$

Other names include:  Box-Tiao distribution (McDonald and Newey 1988), and Power-Exponential (McDonald and Newey 1988, Johnson et al. 1995). Finally, it is worth noting that some economic papers inappropriately ascribe the name 'Subbotin distribution' to an Exponential-Power distribution that has a different functional form.