What size? The last split system I bought cost about AU$1600 installed ( ~$1060) for 3.5kW... Looking at prices now, even going up to a ducted 14kW Mitsubishi Electric heat pump/AC is only AU$5000 (before installation, and you would need to add ducts and vents etc. - that's just the controller, outdoor unit and in-ceiling unit, but still)...

Is this in the US? Maybe the market is distorted just because not enough people are buying them yet, so nobody has much stock and there's little competition... We have that problem I think for air to water - super common in the UK and Europe, but since everybody uses air to air, split and ducted systems are super cheap but air to water are quite expensive. I really would quite like some heated floors in my bathrooms if I end up doing a big renovation, and don't want to go electric in-floor because hydronic with a heat pump should be more efficient.