I did just receive the DC06 Pro and off the bat I agree that it sounds like a supercharged DC03 Pro. It has the same sonic hallmarks: Highly detailed, texture is upfront, propulsive energy, a very fast linear type sound. Like the DC03 Pro, the DC06 Pro does not smooth over harsh production. Listening to Heart's 1985 self titled album, full of great songs with some of the harshest 80s production I can think of, the tizziness from the cymbals and sibilants in voices are evident and not hidden, muted, or smoothed out in any way. When the production is smoother, the DC06 Pro presents it smoothly. This is why I disagree with Andy's description of the DC03 Pro and DC06 Pro as colored. I think both are extremely neutral in that they present the recording without significant changes. The 03 Pro and 06 Pro are undeniably more forward but that is not the same thing as coloration.
Compared to the 03 Pro, the 06 Pro has a clear tighter control over the drivers (in BAL out, I'm not bothering with the SE on this one since all my headphones are balanced capable), and better layering, with more nuance in the presentation of instruments. I am hearing more detail, and also can discern more of the mix in terms of instruments that are mixed higher or lower. Listening to Art Blakely Live in Tunisia, I get a sense of space around the instruments. The soundstage feels close to the head though, not massively expansive, and is wider than it is tall.
I've got a DC04 Pro coming for a direct comparison, but that's shipping from China and with the holidays probably won't have it until January. Without that comparison, I can say the DC06 Pro is everything I like about the DC03 Pro, only more so.
Listening done with IE600s with IE Pro tips and ETA O2, source is an iPhone 15 Pro Max and a Macbook pro.