<s>[INST] Smith Normal forms in a polynomial ring 
Let $M$ be a $\mathbb{C[x]}$ module with generators $m_1,m_2$ and relations : $$(x^2+ix)m_1+(x+i)m_2=0 \\ (-2x+2i)m_1+ (x^2+1)m_2=0 $$
Find integers $t,n_1,...,n_s \in \mathbb{N_0}$ and $\lambda_1,\lambda_2,...,\lambda_s\in \mathbb{C}$ such that the following holds: $$\mathbb{C[x]^t}\oplus\mathbb{C[x]}/((x-\lambda_1)^{n_1})\oplus...\oplus\mathbb{C[x]}/((x-\lambda_s)^{n_s}) \cong M $$

Unless my Work so far is completely wrong I started with the presentation matrix $\begin {pmatrix} x^2+ix & x+i \\-2x+2i & x^2+1\end {pmatrix}$ and by using column/row operations got to the point where $\begin {pmatrix} x+i & 4i \\0 & -x^3-3x+2i\end {pmatrix}$ but I haven't been able to get it into proper Smith normal form so I don't know how to proceed from this point.
 [/INST] You're very close to finishing it. Once you have a unit, like $4i$, you can rescale a row or column to make it $1$ and then begin canceling all the other entries in its row and column.
\begin{align*}
\begin {pmatrix} x+i & 4i \\0 & -x^3-3x+2i\end {pmatrix} &\leadsto
\left(\begin{array}{rr}
4 i & x + i \\
-x^{3} - 3 x + 2 i & 0
\end{array}\right)\\
&\leadsto
\left(\begin{array}{rr}
1 & -\frac{1}{4} i x + \frac{1}{4} \\
-x^{3} - 3 x + 2 i & 0
\end{array}\right)\\
&\leadsto
\left(\begin{array}{rr}
1 & -\frac{1}{4} i x + \frac{1}{4} \\
0 & -\frac{1}{4} i x^{4} + \frac{1}{4} x^{3} - \frac{3}{4} i x^{2} + \frac{1}{4} x - \frac{1}{2} i
\end{array}\right)\\
&\leadsto
\left(\begin{array}{rr}
1 & 0 \\
0 & -\frac{1}{4} i x^{4} + \frac{1}{4} x^{3} - \frac{3}{4} i x^{2} + \frac{1}{4} x - \frac{1}{2} i
\end{array}\right)\\
&\leadsto
\left(\begin{array}{rr}
1 & 0 \\
0 & x^{4} + i x^{3} + 3 x^{2} + i x + 2
\end{array}\right)
\end{align*}
</s>