Photos and story from Marty Horowitz 5/23/2017
Marty writes of these Arroyo Bluets: In this sequence, the female (with the male firmly attached in tandem), goes under the water to oviposit among the submerged vegetation. I wasn’t sure whether this was bad intent by the male or something under the surface dragging her down. But, it turned out to be "normal" according to Rich Bailowitz, our local expert who literally wrote the book A Field Guide to Damselflies & Dragonflies of Arizona and Sonora.
Consider yourself lucky to have seen this process. Often the female does not totally submerge herself as appears to be the case with your photos. What happens on species like this is that once the pair is in tandem, the female backs down into the water so that she can deposit the eggs where she wants - on or under or near plant material. The male stays attached throughout the process.
Note that in the photo below, she is completely submerged with only her wings above the surface - amazing!