Can an Enigma-style cipher of sufficient complexity be considered secure today? Regarding the German Enigma machines, if I recall correctly, the reason they were defeated was because the Allies were able to generate a massive database of possible rotor settings, and because the day key was encoded twice in the beginning of each message.

Given that computers of today can simulate an Enigma-ish machine with an arbitrary number and complexity of rotors (for example, a rotor that only shows up every tenth character, or one that goes forward a different number each time), transmission errors are low, and binary data (not plain-text, but gzipped or base64'ed text) tends to beat frequency analysis, would it be possible to use an Enigma style cipher to encrypt data today, or would it be susceptible to similar pitfalls as the original Enigma cipher?

As a more general question, can other older ciphers be "re-invented" for computers and be viable in today's world?
Yes, if you are willing to throw enough resources at it. Only the most fatally flawed schemes cannot be rescued (in practical terms) given enough additional computation. Since you are even willing to enhance the rotor complexity, you could actually use it to implement a modern algorithm exactly. The ability for a rotor to advance "forward a different number each time" makes it sound like quite a programmable architecture.

But why would you want to? (other than it being a cool project of course) What you would end up with would certainly be many times less efficient than a modern cipher running on modern hardware. Additionally, an attacker who was able to develop custom hardware (FPGAs or ASICs) would likely have access to a far more efficient implementation than the defender.