conventions - What is the meaning of the double complex integral notation used in physics?

In Altland and Simons' condensed matter book, complex Gaussian integrals are introduced. Defining $z = x + i y$ and $\bar{z} = x - i y$, the complex integral over $z$ is

$$\int d(\bar{z}, z) = \int_{-\infty}^\infty dx \, dy.$$ In this way, any integral over $z$ can be done by just breaking into real and imaginary parts.

I'm confused about how one would actually use the notation on the left, as it is. It seems it must have some meaning besides just $dx \, dy$, or there should be no point in introducing it.

It is possible to break the double integral $\int d(\bar{z}, z)$ into two single complex integrals and do them individually? For example, if we were to write

$$\int d(\bar{z}, z) = \int d \bar{z} \int dz$$ then what would the bounds of integration be? For the inner integral, isn't the value of $z$ determined by the value of $\bar{z}$? Alternatively, if we regard $z$ and $\bar{z}$ as independent, then where does the constraint $\bar{(z)} = \bar{z}$ come in? Should each of these integrals be regarded as regular integrals or contour integrals? If we don't break the integral into two, is $d(\bar{z}, z)$ some kind of area element? In that case, how do you do a complex surface integral?

Overall I don't understand what object $d(\bar{z}, z)$ is. What is it, and how do we integrate over it?