1) If you're throwing lots of money around, there's a trail of evidence--electronic, paper, gossip. 2) If people know you're trying to hide--because you're getting face change surgery, because you're trying to launder gobs of cash--you're vulnerable to extortion, theft, and murder, in the moment and in perpetuity. (Again, the world is an ugly place filled with untold numbers of nasty people and billions of desperate people.)
You need either the law or a credible threat of retaliation to protect large assets. The former requires identity and a trail of evidence of ownership, not to mention reliable law enforcement; the latter requires reputation, which requires identity and a trail of evidence of retaliation. And except for industrial countries with a strong rule of law (i.e. the places you're probably trying to flee), both the law and the jungle require regular payments in cash, favors, or blood, commensurate with the assets you're trying to protect, to maintain the state of things. Not only can that quickly diminish your wealth, you're creating an ever larger footprint.
Is it possible? Theoretically, just like in the movies. But it's not easy. And it's not easy to pivot from an ultra rich, jet setting, flamboyant con-man to an anonymous nobody. It's worth noting that your link proves my point--his fake identity was readily discovered with a routine search warrant and home inspection. Epstein wasn't willing to give up the comforts of his public persona and the convenience of switching between his identities. And he clearly wasn't willing to flee before he got caught, which is something of a prerequisite.
Having some money is a good start. You can buy a bar or a hostel in some far flung part of the world where nobody asks questions, and hopefully you can coast for several decades. Maybe you can hide just enough gold and currency to maintain a comfortable life without drawing attention; enough to match the flow of retirement benefits (Social Security, etc), which can often be drawn down regardless of residency and are a common source of income for older expats. Otherwise, it's just mo' money, mo' problems.
On the other hand, it's rare for authorities to care enough to hunt someone down to the ends of the earth. But I assume the context here is a motivated intelligence agency or similarly resourceful and persistent organization (e.g. DoJ no longer able to turn a blind eye). Either way, though, the easiest way to hunt someone down is to follow the money.
Sources: Movies. Education. Traveling and meeting "interesting" expats from various countries fleeing everything you'd expect (wives, taxes, poverty, etc) except prosecution for heinous crimes, if their stories are to be believed.