Let $\mathcal{H}$ denote the Hurwitz Quaternion, i.e the subring of the ring of real quaternions that is defined in the following way: $$\mathcal{H} = \{ m_0 \zeta + m_1 i + m_2 j + m_3 k \mid m_i \in \mathbb{Z} \}$$ where $\zeta = \frac{1}{2}(1 + i + j + k)$. Let $N$ be the standard norm defined on the ring of real quaternions. i.e $N(x) = x_0^2 + x_1^2 + x_2^2 + x_3^2$ for $x = x_0 + x_1 i + x_2 j + x_3 k$
The claim is that norm of any non-zero Hurwitz Quaternion is always a positive integer. I am stuck on one piece of the argument. Here is what I have so far:
Let $x$ be an arbitrary element of $\mathcal{H}$ such that $x = m_0 \zeta + m_1 i + m_2 j + m_3 k$ , $m_i \in \mathbb{Z}$. Then we have that $$ x = \frac{m_0}{2} + \bigg{(} \frac{2m_1 + m_0}{2} \bigg{)} i + \bigg{(} \frac{2m_2 + m_0}{2} \bigg{)} j + \bigg{(} \frac{2m_3 + m_0}{2} \bigg{)} k $$ and so \begin{align*} \begin{split} N(x) &= \bigg{(} \frac{m_0}{2} \bigg{)}^2 + \bigg{(} \frac{2m_1 + m_0}{2} \bigg{)}^2 + \bigg{(} \frac{2m_2 + m_0}{2} \bigg{)}^2 + \bigg{(} \frac{2m_3 + m_0}{2} \bigg{)}^2\\\\ &= \frac{m_0^2 + 4m_1^2 + 4m_1m_0 + m_0^2 + 4m_2^2 + 4m_2m_0 + m_0^2 + 4m_3^2 + 4m_3m_0 + m_0^2}{4}\\\\ &= \frac{4(m_0^2 + m_1^2 + m_2^2 + m_3^2 + m_1m_0 + m_2m_0 + m_3m_0)}{4}\\\\ &= m_0^2 + m_1^2 + m_2^2 + m_3^2 + m_1m_0 + m_2m_0 + m_3m_0 \in \mathbb{Z} \end{split} \end{align*}
The last three terms are concerning me. Why can I argue that the sum of $m_1m_0, m_2m_0, m_3m_0$ are going to enable the norm to always be a positive integer?
As you wrote yourself, the norm is the sum of four squares and thus non-negative. If you now further assume that not all the $m_i$ are zero, we have thus a positive integer.