Does alpha need to be adjusted for running Mann-Whitney U test independently for two dependent variables? I am comparing two group means on two dependent variables (aggression and stress levels) using Mann-Whitney U test by SPSS. My question is that do I need to adjust the alpha level for this? 
 A: If you want to control family-wise $\alpha$
If you want to control your family-wise Type I error rate, then you would need to adjust your alpha level. If you wanted to do this a simple option would be to set your per comparison alpha to be your desired family wise alpha divided by the number of significance tests you are running. E.g.,
$$\alpha_{PC} = \frac{\alpha_{FW}}{c}$$
Thus, in your case with $c=2$ tests and a convention $\alpha_{FW}=.05$, then:
$$\alpha_{PC} = \frac{.05}{2} = .025$$
But do you want to control family-wise $\alpha$?
In many cases were separate analyses are run, no attempt is made to adjust $\alpha$. This is particularly true where the hypotheses are small in number.
Decreasing $\alpha$ will decrease your chance of a Type I error if $H_0$ is true, but decrease your statistical power if $H_0$ is false.
Ultimately you are making a judgement about the cost and benefits you assign to Type I and Type II errors.
The answer to this question would be no different if you were asking about doing two t-tests.
More broadly there are also often options for making analyses more parsimonious. For example, people can create composites out of their dependent variables and run one test on that or run MANOVA. That said if they are distinct dependent variables, you may still want to test them as discrete hypotheses.
