Is it a good idea to use the entire Unicode range to generate a random password rather than limited ranges?

This sounds a lot like Fencepost Security. Imagine you're running a facility that has chain-link fencing around it that is 500 feet high. How much would the security improve by making that fencing 3,000 feet high? None - because anyone trying to get in isn't going to climb the 500 feet; they're going to dig underneath, cut a hole, etc.

Likewise, you've got a password that's, say, 20 random alphanumeric characters. That's 62^20 possibilities. You're considering changing it to 20 random unicode characters. Which raises the possibility space much higher, except brute-forcing a 20-character randomized password isn't how things are going to get compromised.

This is a good idea from a security perspective. A password containing unicode characters would be harder to brute-force than a password containing ASCII characters of the same length. This holds up even if you compare byte-length instead of character length, because Unicode uses the most significant bit whereas ASCII does not.

However, I think it wouldn't be practical since Unicode bugs are so common. I think if you use Unicode passwords everywhere you will encounter more than a couple of sites where you would have problems logging in, because the developers didn't correctly implement Unicode support for passwords.

Ignoring the Security by Obscurity argument it is a basic question of entropy. An 8 character unicode password is more secure than an 8 character ASCII password but less secure than a 64 character ASCII password.

In general I agree with Sjoerd - these are likely to cause more inconvenience than benefit. On top of this if ever you need to manually enter a password random Unicode is likely going to make your life miserable.

However for the edge case where you need to use a service which actively supports unicode whilst enforcing a maximum password length limit (again ignoring this usually indicates other security failings) there is an argument for it.