When I study the Bayesian econometrics, the book firstly introduces Gamma-Normal distribution as (conjugate) prior, then the posterior will have the same distribution as the prior. But my question is, why can't we use other simple distribution as prior for the introduction example, such as normal distribution, why do we have to use a such as strange 'compound' distribution?
There is no reason to use any particular prior in a Bayesian analysis. Conjugate priors are chosen out of convenience so that it is easy to express the posterior distribution in a simple form as a member of the same family as the prior. With the introduction of Markov Chain Monte Carlo methods investigators use whatever priors they find to be realistic for the model parameters. Now for an itnroductory course it may be simpler to get the concepts across when it si easy to compute the posterior distribution. Simple priors will not always lead to nice posterior distributions but with conjugate priors you are safe. That is probably why it was done that way.