I'll admit that as of this writing, I'm short on ideas. I hope I've thought of some fresh ones by the time this gets posted, lest my flow break down. To stimulate me, here's a list of the basic fields of science, and ways to make them more strange and deranged than they already are. So each one is prefixed with "mad," because this is that kind of blog.
* Mad Mathematics
According to Wikipedia, math is the study of "quantity, structure, space, relation, change, and various topics of pattern, form and entity." In other words, quantifying and analyzing things. Math makes life's messy generalities into abstract, numeric patterns.
To twist mathematics, one typically winds up with low-rigor, cranky and ill conceived theories. Mad Mathematics is typically wrong, or worse.
* Mad Physics
Physics is a math-ified study of reality and how it works. Events are understood according to equations, which need vigorous and rigorous testing. Experiments involve matter, motion, elemental forces, machines, and quantum events. Engineering involves a heavy amount of physics, as machines built in accordance with the rules of reality work better, and machines built contrary to the rules of reality don't work at all.
Mad Physics would involve strange combinations of the above, often in impractical ways. Quantum Nuclear Hammer Propulsion! Surely, you won't mind if I do it remotely from a vast distance, as I like being alive too much to be sure if it won't kill me if I do it in person.
* Mad Chemistry
Chemistry involves more detailed studies of matter, what it's made of, and the effects of combining it. New materials, pharmaceuticals, and engineering techniques are the direct results of chemistry experiments.
Mad chemistry would involve combinations suspected to be dangerous.
Again, preferably remotely, as I don't see a point in being killed for science when I can endanger a probe and a camera instead. Probe 'n' camera are cheaper than my own flesh.
* Mad Biology
Biologists study life. From the smallest cell components to the behavior of organisms. It dovetails with chemistry ("Biochemistry") because life is made of stuff, with computer science for the information of life like DNA and behavior patterns, and with a few other disciplines.
Mad biology would involve strange life, either spliced from existing plants, animals, and bacteria into forms that probably would not evolve naturally, or outright completely artificial (which is harder than it sounds). I wonder if I could invent something with nipples for eyes! (Hint: Technically it's possible, but it would be blind and not inherently useful. If the nipples could provide milk, it's still limited by the entire organism's metabolism.)
* Mad Psychology
Psychologists study minds. Both for analytic (Why do people buy FLIBB brand cola and not YUPTA brand cola?) and therapeutic (It's unfortunate that your mom killed your dad with an eggbeater while you were watching, now let's talk about your fear of egg beaters) reasons.
Mad Psychology would probably venture into the unethical, (Convince people of things that are blatantly untrue! Cause insanity in your patients!) the blatantly false, (behold my obviously wrong theory of personal development!) or the unspeakably bizarre (What is the purpose of love? I think I'll find out by torturing monkeys).
* Mad Computer Science
Computer science studies how to store and manipulate information, how to automate the storing and manipulation of information, and to keep the complexity to a minimum. Machines might be powerful at math and logic, but they're not omnipotent, and some things that are easy for us are mind-bogglingly difficult for them. Take vision, for instance. It's easy for us to know what we're looking at, but give a computer a camera and it takes zillions of operations, and even then it's not entirely sure.
Mad computer science might involve unconventional programs, odd storage methods, or truly crazy automation.
* Mad Economics
Economics has often been called the "Dismal science" because many implications profoundly bother people. It also is uncomfortably between the "hard" sciences where there is one objective truth, and the "soft" sciences where many opinions prevail at once. (Some parts of economics are definitely true, but others are up to interpretation. It frustrates economists.) Economics studies money and value, in an attempt to set up society to maximize both wealth and fairness.
Mad economics would take unconventional theories on psychology, value, or production. Special care would need to be taken to avoid being wrong. (or worse.)
* Mad Engineering
Engineers make machines to solve problems. Mad engineers make strange machines to solve conventional, strange, or nonexistent problems.
* Mad Sociology
Sociology is like mass psychology. Studying the beliefs and practices of entire societies and cultures. Since it's in aggregate, it's very confusing.
Sociology is already so strange that mad sociology boggles my mind even attempting to imagine it.
* Mad Astronomy
In pre-scientific times, many cultures believed that the stars held answers to daily riddles, and predicted the future. So they studied them, with a field they called Astrology. Later, people realized that the stars couldn't possibly know the future, but still found stars interesting, so they looked at them for purely theoretical reasons. (Astrology would mean "star study," so "Astronomy" is "star naming." They just look and name.)
So mad astronomy ... would be giving stars strange and hilarious names. Check out the brightness of Glimglomyeeeyeeboing. (Which we named because we can't stop sniggering every time we say it.)
* Mad Political Science
"Who gets what and how." Theories on power, law, diplomacy. Good and bad ways to run a country.
I'm sure you can rule in a strange way that isn't technically worse than what's done now. One could always advocate ruling systems that have yet to be established, like technocracy (rule by a scientific elite), demarchy (rule by randomly selected people), kritarcy (rule by judges, according to principles), Gerontocracy(rule by older people), or Ochlocracy (rule by whoever shows up today). Most of these are obviously bad ideas.
* Mad Electronics
Electronics studies circuits, and how to make electricity do useful work. Including computers, which are made of extra-complicated electronic circuits.
Mad electronics would involve circuit patterns that are ludicrous and make no sense.
* Mad Archeology
Archeology is all about digging up past things to discover what went on there. There's a very established pattern as to the best way to dig to avoid damaging evidence or distorting results.
Mad Archeology is impossible. Deviate from the theory and you've destroyed your own results.
* Mad Dentistry
Dentistry involves keeping human teeth healthy. Teeth need to be cleaned, gums massaged, and if teeth are damaged, repaired. If teeth are knocked out, they need to be kept alive until re-implanted, or they will die and be rejected. Missing teeth can be replaced with prosthetics.
Mad dentistry would involve unconventional techniques. Strange ways of cleaning (ultrasonic waves? Lasers?), strange implants (ceramic bone? Plastic?), and strange repair.
* Mad Geology
How do Earth-bound systems work? Study volcanoes, earthquake, rocks, soil, and all things related to the planet and its workings.
Mad geology would attempt to bend the earth to your own will, perhaps by channeling lava to achieve objectives. (Warning: Lava is dangerous due to its immense heat. Do not attempt to directly touch lava. Do not be in the same room as lava. Do not be within the same mile as lava if you can help it.)
* Mad Medicine
Medicine attempts to maintain human health. Cure and prevent disease, treat accidents, and make the unhealthy healthy.
Mad medicine would involve doing strange things to improve health. Weird surgery, odd chemicals. Test on animals first, because few things are worse than unnecessary surgery.
* Mad Educational Science
How do you teach people things, and get them to remember it? As time goes on, the need for education grows. Society becomes more complicated, and citizens are expected to know more and more. In the middle ages, people were full grown adults at the age of 12, because they didn't need to know very much, and society was simple. By 1900, people had to learn in schools until the age of 18 to do anything productive. Now, people often aren't ready for their ultimate careers until they are 25 or 30. Also, they need to not forget what they learned.
Mad educational science would teach people things in strange ways. Maybe by simulation, game, scent, or direct-brain-stimulation a la "The Matrix."
* Mad Agriculture
People need plants. To eat, to make into clothing, to decorate, and to feed to animals that they then eat. Someone has to grow these plants. And if they're eating animals, someone has to feed and take care of the animals, then kill and process them. (The raising and killing need not be done by the same person.)
Mad Agriculture would grow unusual plants, and/or in unusual ways. Muahahaha, my genetically engineered okra/sweet-potato/tomato will feed billions on scarcely any water! Muahahahaha!
How do languages work?
Mad Linguistics would, I suppose, invent new languages in the name of efficiency. They could have features not existing in any language, like perfect regularity (Esperanto), social engineering (Lojban supposedly makes ones thoughts perfectly logical), or some other features.
What happened in the past?
Like Archeology, it's not safe to meddle with this without sinking into revisionism, crankdom, or some other undesirable result.
* Mad Civil Engineering
People like living in big cities to reduce their need to travel, because they get higher paying jobs, and so on. Cities can easily become miserable hellholes due to their enormous crowds of people. A lot of work is needed to make them nice.
Mad civil engineering would involve strange city plans. Strange features, strange transportation, strange everything.
Architects design buildings. Unless they're landscape architects, in which case they design the space around buildings, like yards. Architecture benefited significantly when people had some understanding of physics and could rely on certain principles to skip the "Oh hey, let's build it and see if it falls down" stage. A good architect is both an artist and a scientist, making a building that both looks wonderful and works excellently for its purpose.
Mad architects would design strange, unconventional buildings that technically fulfilled their purpose, but what the hell?
* Mad Military Science
When a nation wages war, it aims to win. So does the other side. How do you insure that you win?
Mad Military Science would make unconventional choices.