I'm currently using a red cypress chip on my garden. Does anyone know what their effects would be on depleting the soil and such? I've tried Dr Google but of course all I get is advertising.
If you're concerned about initial nitrogen depletion, then add a layer of compost beneath the chips first - composted manure or good garden compost, that sort of thing. Over time, the chips will add to the humus content of the soil, but there is an initial loss of nitrogen in the surface layer of soil for a few weeks.
In regard to its being dyed, so long as the producers have observed the rules about what's in the bag being exactly as described, then the contents should only contain cypress chips, and not questionable woods from other sources, such as those containing arsenical compounds, which are often a component of red dyed woodchips. Red dye may leach onto surrounding hard surfaces, so take care not to get it too close to any paving or other hard landscaping. Unless it states the dye is of vegetable origin, it will contain iron oxide which will be absorbed into the soil over time, but in such tiny quantities it should not have a detrimental effect; it will, though, cause staining on paved surfaces.
Because the product you've linked to is an Australian one, I assume you must be in Australia, which means your weather must be hotting up - you should note that there is a fire risk with wood chip mulch, and guidance on that is given here http://www.live-safe.org/2010/05/06/mulch-fires-common-in-the-landscape/