I'm not sure what you mean by "as advanced as Egyptian medicine." The Ebers Papyrus recommends a poultice of chopped bat for an itchy neck, while the Badianus Manuscript recommends a drink using the ground tail of a dog to cure urinary retention, so there's some overlap in using processed animal parts for no good reason. Coincidentally, a remedy for a child not producing urine from Ebers is to boil an old book in oil, which is then rubbed on the belly.

The problem with pre-modern medicine is that for every prescription which has some value by modern standards (use of honey for poultices and coughs), there are nonsensical concoctions like the above. Then there are the cures which kind of make sense, like the Ebers cure for a crying baby, which is a draft of opium mixed with flyspeck. Remedies in the past were a mixture of practice, tradition, and mysticism, with no small amount of placebo effect, all wrapped up in a package far removed from "evidence-based medicine." Sometimes there were common sense approaches and sometimes there are some surprisingly good recommendations at the core of past prescriptions, but pre-modern medicine was very much an artifact of its particular culture as much as it was a curative regime. The medicinal schema of past societies were necessarily limited by their gaps in their anatomical knowledge, poor understanding of physiology, and general ignorance of the disease etiology. So it's a bit odd to think of one system (particularly something as broad as "Egyptian") as more advanced than any other, when they all faced similar limitations.

Enough pontificating though, let's talk Aztec.

The general Nahuatl term for a physician was ticitl, though there were who might focus on a particular body part (e.g., the eye), as well as midwives. There was also overlap with more esoteric practitioners of astrology and divinination. Sahagún's Historia General (Book 10) describes a ticitl as:

a knower of herbs, of roots, of trees, of stones; she is experienced in these. She is one who has the results of examinations; she is a woman of experience, of trust, of professional skill: a counselor.

The good physician is a restorer. a provider of health, a reviver, a relaxer -- one who makes people feel well, who envelops them in ashes. She cures people; she provides them health; she lances them, she bleeds them -- bleeds them in various places, pierces them with an obsidian lancet. She gives them potions, purges them, gives them medicine. She cures disorders of the anus. She anoints them; she rubs, she massages them. She provides them splints; she sets their bones -- she sets a number of bones. She makes incisions, treats one's festering, one's gout, one's eyes. She cuts growths from one's eyes.

You'll note the female pronoun is used. This is because a ticitl was not necessarily a gendered profession, although the stereotypical "woman physician" was an older woman past her child-bearing years.

In that same book there's also an exhaustive list of gross anatomical terms which also contains some interesting insights into recognized symptoms. For instance the section on the face includes a term on appearing jaundiced (camaoa), while the section on the torso has a label for when it "becomes distend of navel" (xicuitoliui), a classic sign of kwashiorkor.

Sahagún then rounds out by listing a multitude of illnesses categorized by the part of the body they affect, as well as recommended treatments. So an infected ear (nacazqualiztli) could be treated by the application of drops of coyolxochitl sap mixed with chili, three time during the day and three times at night. You might also use saltwater drops mixed with seashell scrapings. Or perhaps you have a tongue abscess. That could be treated by lancing the abscess to remove the blood/pus, and then inserting a salted thread and drinking an infusion of iztac quauitl wood. Or perhaps your nose has been cut off. Well then it would be sutured back on and bathed with salted honey.

These are just some of the simpler remedies listed. There's also more complicated instructions, like the care of a broken leg using splints, bleeding swollen of the foot, and applying a particular poultice. There's also instructions for curing diarrhea using an herbal tea with different additives depending on whether it was being given to an adult or a child.

The Badianus Manuscript I mentioned earlier is likewise a collection of prescriptions for various ailments. Some being quite complex, and some being bluntly simple. A simple drink of new deer horn ground into fermented maguey syrup, for instance, is recommended to ward of lice. Lesions of the feet, meanwhile, call for a multi-day process of soaking the feet in a hot water infusion of a dozen different herbs and crushed stones, followed by adding an ointment to the fire keeping the water hot, and then adding even more herbs to the soak.

The point of all this is that the Aztec world possessed an astounding pharmacopoeia with which to treat a broad array of recognized ailments. It was also flexible in adapting and adopting the new traditions being introduced by Europeans. Both the Historia and Badianus were written in the mid-16th Century and show clear signs on integration of the two cultures. The latter book occasionally references Afro-Eurasian plants/materials (and amusingly gives a cure for a "goat-like smell of the armpits") and the amount of work devoted to parsing the native/non-native influences on Sahagún could easily float or sink a boat. Nevertheless, these were introductions into a sophisticated system of medical practice and theory, one which readily added new techniques and ingredients to its toolset.

This topic has an abundance of nuances to cover, like the role of certain gods associated with particular illnesses (e.g., Tlaloc, the rain god, being associated with pulmonary disease, naturally), the role of the Aztc tripartite soul (tonalli/teyolia/ihiyotl), divine and non-divine etiologies of disease, and even the debate over the hot-cold dichotomy which continues to dominate the practice of curanderos was autochthonous or introduced (or a syncretic system). If you have further questions, please ask.

For some suggested reading, Guerra (1966) is dated, but still a classic overview of Aztec medical praxis. Ortiz de Montellano's (1990) Aztec Medicine, Health, and Nutrition is another recommended read.