Inspirations - The Aztecs

Another story that has always tended to be known by just the one title. In fact, the "naming controversy" is only really confined to the first two stories.
The writer is once again John Lucarotti. He lived for some time in Mexico, so would have become interested in the Aztecs and their culture during that time.
The last historical story had featured real people - Marco Polo and Kublai Khan. Lucarotti could have gone down the same route and featured historical characters - Pizarro and Montezuma II - and shown the actual events surrounding the meeting of the Conquistadors and the Aztecs. The clash of cultures and resulting conflict would seem to have been a more obvious starting point.
Peter Shaffer had used this for his play about the meeting of the Spanish and the Incas - The Royal Hunt of the Sun - which opened in London after The Aztecs was commissioned, but before it was broadcast. The publicity for the play would have helped the Doctor Who story, even though they deal with the Spanish clashing with entirely different indigenous cultures.
Lucarotti instead decides to show us the Aztecs at their peak, before the coming of the Spanish. The arrival of the Europeans is talked about, as something that will happen soon and sweep all of this away.
The two aspects of Aztec culture we know best about - their knowledge and their use of human sacrifice - are embodied in the two main characters whom the time travellers meet. Autloc, High Priest of Knowledge, is presented as a good, kind man, who is open minded. Tlotoxl - High Priest of Sacrifice - is presented as an evil person, an old fashioned villain who is unwilling to countenance that his religion is wrong in any way. For inspiration in his performance, John Ringham looks to Richard III - specifically as portrayed by Laurence Olivier in the 1955 film of Shakespeare's play.

Unusually, Tlotoxl does not get his comeuppance at the conclusion. Autloc wanders off into the wilderness to become a hermit, and Tlotoxl ends up stronger, with his own candidate for High Priest of Knowledge ready to take over.
Setting aside the performance, is Tlotoxl bad? One of the points which the Doctor strives to make in his debates with Barbara is that she cannot judge the Aztec culture by her 20th Century English viewpoint. He points out that in this city, it is Autloc who is the odd man out. Human sacrifice has a role in this society and is accepted. It honours the gods and benefits the society, making the crops grow and the rains fall. Tlotoxl believes this to be true, and so when he suspects that Barbara is a false god he seeks to undermine her by any means possible - including poisoning her. She poses a threat to their established order, so is he wrong for trying to do this? Of course not. We, the viewers, are also looking at the Aztecs filtered through those 20th Century western values.
Barbara wants to cherry pick the good things, as she sees them, of Aztec culture and eliminate the nasty ones - hoping that when Cortes gets here the civilisation will have an easier time. Naive thinking for someone who claims to have specialist knowledge of this period, as European illnesses alone will kill millions, and the Catholic Church will not tolerate any heathen religion, human sacrifice or not. At this point the Church is burning fellow Christians for even the slightest taint of unorthodoxy. The Conquistadors have come for gold and for souls, and that's it.
It should also be noted that the Aztecs weren't just defeated by the Spanish alone. Cortes was aided by thousands of other Mexican troops from neighbouring kingdoms, who wanted to smash Aztec dominance for their own reasons.

The Aztecs is the first story to look at the consequences of time travel. More recently, we have been introduced to the notion of Fixed Points in Time, but this wasn't the case back in 1964. Right from the earliest days of the series, viewers began questioning why it was okay to meddle in the affairs of Skaro, but not of Earth. Shouldn't the Doctor be an observer, who doesn't do anything to upset the order of things wherever he landed? Why was it okay to help the Thals and wipe out the Daleks, but not to save the Aztecs from near genocide?
Story Editor David Whitaker, responding to a letter from a viewer, claimed that History was like a road running across an undulating landscape. At times, the road dipped out of view, so anything could happen, but it had to resume its course after a while. He described History like Justice - not only being done but seen to be done. What we know to have happened always needs to be seen to happen just as the History books tell it. In the next season, Whitaker's successor will have his own ideas about History, and one story will have the time travellers actively trying to prevent something that is seen to have already happened - and they will succeed.
The Doctor has always had a special relationship with the Earth, and we know that one day the Universe will be populated by what look and sound like British people. Maybe this is why Earth history is protected so much - because humans will have such an impact on the cosmos further down the line. The Doctor may always have seen the Fixed Points in Time, just never mentioned them, and so knows that his actions on Skaro, or Vortis, or Peladon, were simply the right thing to do.

A few final points. Lucarotti has done his homework, obviously, so Barbara gets to talk about their gods, and Susan is tutored in some of their ways. The Doctor has to fashion a wheel and pulley system, as the Aztecs did not exploit the wheel.
The Aztecs never used that name to describe themselves - they were the Mexica. It was the Spanish who called them Aztecs.
Cocoa beans were used for barter and as a form of currency. Vassal kingdoms were expected to provide cocoa beans by way of annual tribute. The oily layer which formed on chocolate beverages was also used as a form of sun block. Preparing a cocoa drink wasn't especially used as a form of marriage proposal, but it did form part of the marriage ceremony itself, so Cameca might be jumping the gun a bit here.
The programme came in for some criticism about the authenticity of the costumes. They were all properly researched. The location of this city is never specified, and might lie at a high altitude, and Mexico does have its seasons like everywhere else - so Tlotoxl and company might not necessarily be over-dressed.
The Aztecs inherited a great deal of astronomical knowledge from earlier Mesoamerican cultures, and so would have been able to predict a solar eclipse. Their cities were laid out on astronomical alignments.
Of the character names, Ixta could derive from the Mexican coastal town of Ixtapa, but it is also another name for Iztaccihuatl - the country's third highest mountain. Iztapalapa is a suburb of Mexico City, its most densely inhabited.
Next time, we head off into Space, in the 28th Century, and it looks like the TARDIS has never landed on a spaceship before...