Bjarni Jonsson had a result somewhat like the one mentioned by Ricky Demer.
It appears in
Jónsson, Bjarni On direct decompositions of torsion-free abelian groups.
Math. Scand. 5 1957, 230-235.
I will copy the description of his group here.
Consider a four dimensional vector space $V$ over the field of rational numbers, let $\{x, y, z, u\}$ be a basis for $V$, and let
\[x' = 3x-y\quad \textrm{and}\quad y' = 2x-y.\]
Then $\{x', y', z, u\}$ is also a basis for $V$, and
\[x = x'-y' \quad \textrm{and}\quad y = 2x'-3y'.\]
Let $A, B, C, D$ consist of all those elements of $V$ which can be written in the forms
\[\begin{array}{rl}
\frac{a}{5^n}x, \quad &\frac{b}{5^n}y+\frac{c}{7^n}z+\frac{d}{11^n}u+
\frac{e}{3}(y+z)+\frac{f}{2}(y+u),\\
\frac{a}{5^n}x'+\frac{c}{7^n}z+\frac{e}{3}(x'-z),
\quad &\frac{b}{5^n}y'+\frac{d}{11^n}u+\frac{f}{2}(y'-u),
\end{array}
\]
respectively, with $a, b, c, d, e, f$ and $n$ being integers.
Jonsson proves that $A\oplus B = C\oplus D$, that $A, C, D$
are indecomposable (he states without proof that the same
is true for $B$), and that $A$ is the only one of them
that is embeddable in $\mathbb Q$.
$\ldots$ I particularly welcome examples with a simple description $\ldots$
Maybe Jonsson's example does not satisfy this criterion.
If you only need it for illustration purposes, here is an
easy example from a category of modules.
Let $R$ be an integral domain that has comaximal ideals $I$ and $J$
that are not principal. For example, you could take
$R=\mathbb C[x,y]$, $I = (x,y)$, $J = (x-1,y)$. The exact sequence of $R$-modules
$$
0\to I\cap J\to I\oplus J\stackrel{\alpha}{\to} R\to 0, \quad\alpha(i,j)=i+j,
$$
splits because the right term is free.
Hence $I\oplus J\cong R\oplus (I\cap J)$.
Each of $R, I, J, I\cap J$ is indecomposable, since they are
ideals in a domain. $R$ is cyclic, but $I$ and $J$ are not,
so at least one term on the left is not isomorphic to at least one term on the right.
