Background: In my senior year of undergrad, I was a TA for our school's "Fractal Geometry" course, having worked with the professor before on a research project in fractals. We followed some of the materials provided (publicly!) in Yale's Fractal Geometry course.

There was a particular focus on Iterated Function Systems, which are great ways to generate fractals of "type 1", per OP's list. (Indeed, the Yale course was created by Michael Frame, who was a collaborator on our research into IFS.) There are some great visuals and applets provided to explore the main ideas and appeal to some more visual/hands-on learners. I highly recommend at least looking through the course's contents, whether or not you end up using all of it.

If you are looking to address a student audience of senior math majors, you might want to hone in on some of the following sections. Note: This list is certainly not exhaustive; this is just what I found scanning through now. But these will at least require the students to use some of their math knowledge and maturity.

• Algebra of Dimensions: Does knowing the dimension of $A$ and $B$ tell us anything about the dimensions of $A\cap B$, $A\cup B$, $A\times B$, $A$ projected onto $B$?
• Combinatorics of the Mandelbrot set
• Newton's Method for complex numbers, followed by a use of such a method on the Mandelbrot set
• Various topics in the chapter on Chaos

In addition, there some appendix resources:

• Lesson plans and a list of required mathematical topics, sorted by fractal topic
• "Lab" activities for exploring certain topics (could serve as in-class activities or take-home projects)

Regarding your question about whether to require complex analysis: You don't have to. This will depend on how much material on the Mandelbrot/Julia sets you want to cover, and to what depth you do so. I would recommend tending towards the side of not requiring much familiarity with complex analysis, and only a working knowledge of the algebra of complex numbers; hopefully, this will make more students want to take this course!

Background: In my senior year of undergrad, I was a TA for our school's "Fractal Geometry" course, having worked with the professor before on a research project in fractals. We followed some of the materials provided (publicly!) in Yale's Fractal Geometry course.

There was a particular focus on Iterated Function Systems, which are great ways to generate fractals of "type 1", per OP's list. (Indeed, the Yale course was created by Michael Frame, who was a collaborator on our research into IFS.) There are some great visuals and applets provided to explore the main ideas and appeal to some more visual/hands-on learners. I highly recommend at least looking through the course's contents, whether or not you end up using all of it.

If you are looking to address a student audience of senior math majors, you might want to home in on some of the following sections. Note: This list is certainly not exhaustive; this is just what I found scanning through now. But these will at least require the students to use some of their math knowledge and maturity.

• Algebra of Dimensions: Does knowing the dimension of $A$ and $B$ tell us anything about the dimensions of $A\cap B$, $A\cup B$, $A\times B$, $A$ projected onto $B$?
• Combinatorics of the Mandelbrot set
• Newton's Method for complex numbers, followed by a use of such a method on the Mandelbrot set
• Various topics in the chapter on Chaos

In addition, there some appendix resources:

• Lesson plans and a list of required mathematical topics, sorted by fractal topic
• "Lab" activities for exploring certain topics (could serve as in-class activities or take-home projects)

Regarding your question about whether to require complex analysis: You don't have to. This will depend on how much material on the Mandelbrot/Julia sets you want to cover, and to what depth you do so. I would recommend tending towards the side of not requiring much familiarity with complex analysis, and only a working knowledge of the algebra of complex numbers; hopefully, this will make more students want to take this course!