Basic Algebras: Definition A $k$-algebra $A$ is called basic if for every set of primitive orthogonal idempotents $\left\{e_1, \dots , e_n\right\}$ such that $1=\sum_{i=1}^ne_i$ we have that $$e_iA\cong e_jA\Leftrightarrow i=j.$$
If $A$ is a finite-dimensional $k$-algebra such that we have a decomposition $1=\sum_{i=1}^ne_i$ with $e_iA\cong e_jA\Leftrightarrow i=j$, then by the Krull-Schmidt theorem we have that if $1=\sum_{i=1}^ne_i'$ is another such decomposition, then $e_i'A\cong e_j'A\Leftrightarrow i=j$. Hence for finite-dimensional algebras the definition of a basic algebra can be formulated in terms of one decomposition of $1$ into primitive orthogonal idempotents.

But is there an example of an (infinite dimensional) algebra with two such decompositions such that one looks basic and the other doesn't? 

 A: Let $B=M_2(k)$ be the algebra of $2\times 2$ matrices, with $1_B=e_B+f_B$ a decomposition into primitive orthogonal idempotents, so $e_B$ and $f_B$ are conjugate.
Let $C=k\times k$, with $1_C=e_C+f_C$ a decomposition into primitive orthogonal idempotents, so $e_C$ and $f_C$ are not conjugate.
Let $A$ be the coproduct of $A$ and $B$ in the category of $k$-algebras (i.e., the $k$-algebra generated by $B$ and $C$ subject to no relations other than those that already hold in $B$ and $C$). Concretely, if you choose bases $\{1_B,b_1,b_2,b_3\}$ and $\{1_C,c\}$ of $B$ and $C$, then $A$ has basis the set of words in $\{b_1,b_2,b_3,c\}$ with alternating $b$s and $c$s.
Then $A=e_BA\oplus f_BA$ where $e_BA\cong f_BA$, and $A=e_CA\oplus f_CA$ where $e_CA\not\cong f_CA$.
The idempotents all remain primitive in $A$, since if, for example, $e_B$ decomposes as $e_B=e+f$, then $e_B$ commutes with $e$ and $f$, but it's easy to check that an element of $B$ can only commute in $A$ with other elements of $B$, so this decomposition would already happen in $B$.
