Are there more modern password hashing methods than bcrypt and scrypt?

In cryptography, "new" is not synonymous to "good". That bcrypt is twelve years old (12 years... is that really "old" ?) just means that it sustained 12 years of public exposure and wide usage without being broken, so it must be quite robust. By definition, a "newer" method cannot boast as much. As a cryptographer, I would say that 12 years old is just about the right age, and anything younger than, say, 5 years, is definitely "too young" for general deployment (of course, these estimates depend on how much exposure the algorithm got; an early, wide deployment, although risky for those who decide to deploy, will go a long way toward building confidence in security -- or revealing weaknesses at an early stage).

Scrypt is much newer than bcrypt; it dates from 2009. The idea is quite smart. Namely, slow password processing is meant to make dictionary attacks N times more expensive for the attacker, while implying that normal processing is N' times more expensive for the honest systems. Ideally, N = N'; the scrypt author argues that with PBKDF2 or bcrypt, use of ASIC allow an attacker to get a N much lower than N' (in other words, the attacker can use specialized hardware, because he is interested only in breaking a password, and thus hashes many more passwords per second and per spent dollar than the honest system). To fix that, scrypt relies on an algorithm which requires quite some RAM, since fast access RAM is the specialty of the PC, and a sore point of ASIC design. To which extent scrypt is successful in that area remains to be measured; 2009 is recent times, and the figures given by the scrypt author are based on 130 nm ASIC technology and an hypothesis of "5 seconds worth of processing", which is quite beyond what the average user is ready to wait.

For practical usage now, I recommend bcrypt.

Scrypt notwithstanding, current research on the concept of password processing is more about specialized transforms that allow more than mere password verification. For instance, the SRP protocol allows for a cryptographic key agreement with mutual password-based authentication, and resilient to dictionary attacks (even in the case of an attacker actively impersonating the client or the server); this calls for a bit of mathematical structure, and the password-hashing in SRP involves modular exponentiation.

It's 2016, so it's well worth revisiting this 5 year-old question. There was a Password Hashing Competition conducted from 2013 to 2016, which accepted 24 submissions and selected Argon2 as its recommended password hashing algorithm.

Everything that Thomas said about new vs. good still applies. As recently as February 2016 (after the end of the contest) Argon2 had a small change made to it (version 1.3) in order to harden it against a minor weakness. So I wouldn't automatically jump on it just yet, but it's work keeping it in the corner of one's eye, since it's likely that Argon2 will become more common in the coming years.