a mathematical object that becomes zero when squared 1) is there a math object or number (any type; quaternion or its further generalization, for example) that becomes zero when some operation is applied to the object using itself? The object must not be zero.
also, when does such object not become zero when operated on it by another object?
2) is there an object that becomes zero when operated(multiplied) by another object?
3) is there any math object or matrix that satisfies the following property (I ask for the object that satisfies both constraints):
a)$ab+ba=0$ where a and b are math objects or matrices
b) $A^2=B^2$
for all objects I assume nonzero.
 A: *

*Is there such a quaternion: no. Since nonzero quaternions have inverses, $a^2=0$ would imply $a=0$, after both sides are multiplied by $a^{-1}$. But as for generalizations, rings in general provide plenty of examples. For instance, in $\mathbb{Z}/4\mathbb{Z}$, you have $(2+4\mathbb{Z})^2=0+4\mathbb{Z}$. You can also do it with matrices: $\begin{bmatrix}0&1\\0&0\end{bmatrix}$ squares to zero.

*In quaternions, again no: if $ab=0$ for nonzero $a$ and $b$, multiplying both sides by $a^{-1}$ gives you $b=0$, a contradiction. But again, general rings provide thousands of examples. Both of the examples from item 1 work, for example.

*As Arthur already noted in a comment, $ab=-ba$ is equivalent to your 3a, and we say that $a$ and $b$ "anticommute." He gave an example of a nonassociative algebra where that happens for many elements, but there are many associative algebras that do that too. I will give you a matrix example: $\begin{bmatrix}0&1\\1&0\end{bmatrix}\begin{bmatrix}2&3\\-3&-2\end{bmatrix}=-\begin{bmatrix}2&3\\-3&-2\end{bmatrix}\begin{bmatrix}0&1\\1&0\end{bmatrix}$. For 3b in the case of the quaternions, I will let you try as an exercise that if $a^2=b^2$, then $a=\pm b$. Again for general rings it is not so easy, and there are as many examples as you want. For example, $\begin{bmatrix}1&0\\0&1\end{bmatrix}^2=\begin{bmatrix}0&1\\1&0\end{bmatrix}^2$.
