I've noticed Apple screens (desktop and ipads) and recent laptops (including non Apple) produce better picture than any LCD monitors I can find. What kind of monitors should I look for that have same picture quality?
TL;DR: the Dell P2415Q
Given the target of a 24" 4K monitor, we're basically left with two options: the LG 24UD58-B and the Dell P2415Q.
Both use the same IPS panel manufactured by LG; the differences are in out-of-the-box calibration (Dell has a better reputation and provides statistics in the box for each monitor), connectivity (the 24UD58-B has 1x DisplayPort and 2x HDMI while the P2415Q has 3x DisplayPort, 1x HDMI, and 1x Mini DisplayPort), and Freesync (only the 24UD58-B has Freesync.)
Because you don't game on an AMD card at the moment, and live in Australia where availability of the 24UD58-B is quite poor (I couldn't find any listings at the moment), this leaves the P2415Q, which just squeaks in under the budget at $585 AUD.
Here's a full review of the monitor. Pros are excellent calibration, good contrast (for an LCD), good pixel response times, and of course, 4K. Weaknesses were confined to a too-narrow color space and a heavily matte finish.